meskhi a. measure of non-compactness for integral operators in weighted lebesgue spaces (nova,...

7/23/2019 Meskhi a. Measure of Non-compactness for Integral Operators in Weighted Lebesgue Spaces (Nova, 2009)(ISBN 1…

http://slidepdf.com/reader/full/meskhi-a-measure-of-non-compactness-for-integral-operators-in-weighted-lebesgue 1/135



MEASURE OF NON-COMPACTNESS FOR

INTEGRAL OPERATORS IN WEIGHTED

LEBESGUE SPACES

No part of this digital document may be reproduced, stored in a retrieval system or transmitted in any form or

by any means. The publisher has taken reasonable care in the preparation of this digital document, but makes no

expressed or implied warranty of any kind and assumes no responsibility for any errors or omissions. No

liability is assumed for incidental or consequential damages in connection with or arising out of information

contained herein. This digital document is sold with the clear understanding that the publisher is not engaged in

rendering legal, medical or any other professional services.



MEASURE OF NON-COMPACTNESS FOR

INTEGRAL OPERATORS IN WEIGHTED

LEBESGUE SPACES

ALEXANDER MESKHI

Nova Science Publishers, Inc.

New York



c 2009 by Nova Science Publishers, Inc.

All rights reserved. No part of this book may be reproduced, stored in a retrieval system or

transmitted in any form or by any means: electronic, electrostatic, magnetic, tape, mechanical

photocopying, recording or otherwise without the written permission of the Publisher.

For permission to use material from this book please contact us:

Telephone 631-231-7269; Fax 631-231-8175

Web Site: http://www.novapublishers.com

NOTICE TO THE READER

The Publisher has taken reasonable care in the preparation of this book, but makes no expressed or

implied warranty of any kind and assumes no responsibility for any errors or omissions. Noliability is assumed for incidental or consequential damages in connection with or arising out of

information contained in this book. The Publisher shall not be liable for any special, consequential,

or exemplary damages resulting, in whole or in part, from the readers’ use of, or reliance upon, this

material.

Independent verification should be sought for any data, advice or recommendations contained in

this book. In addition, no responsibility is assumed by the publisher for any injury and/or damage

to persons or property arising from any methods, products, instructions, ideas or otherwise

contained in this publication.

This publication is designed to provide accurate and authoritative information with regard to thesubject matter cover herein. It is sold with the clear understanding that the Publisher is not engaged

in rendering legal or any other professional services. If legal, medical or any other expert assistance

is required, the services of a competent person should be sought. FROM A DECLARATION OF

PARTICIPANTS JOINTLY ADOPTED BY A COMMITTEE OF THE AMERICAN BAR

ASSOCIATION AND A COMMITTEE OF PUBLISHERS.

Library of Congress Cataloging-in-Publication Data

Available upon request.

ISBN 978-161728-536-3 (E-Book)

Published by Nova Science Publishers, Inc. New York



Contents

Preface vii

Basic Notation xi

1 Basic Ingredients 11.1. Homogeneous Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

1.2. Measure of Non–compactness . . . . . . . . . . . . . . . . . . . . . . . . 3

1.3. Hardy–type Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4. L p( x) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

1.5. Schatten–von Neumann Ideals . . . . . . . . . . . . . . . . . . . . . . . . 22

1.6. Singular Integrals in Weighted Lebesgue Spaces . . . . . . . . . . . . . . . 23

1.7. Notes and Comments on Chapter 1 . . . . . . . . . . . . . . . . . . . . . . 25

2 Maximal Operators 272.1. Maximal Functions on Euclidean Spaces . . . . . . . . . . . . . . . . . . . 27

2.2. One–sided Maximal Functions . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3. Maximal Operator on Homogeneous Groups . . . . . . . . . . . . . . . . . 34


3 Kernel Operators on Cones 37

3.1. Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.2. Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3.3. Schatten–von Neumann norm Estimates . . . . . . . . . . . . . . . . . . . 45

3.4. Measure of Non–compactness . . . . . . . . . . . . . . . . . . . . . . . . 473.5. Convolution–type Operators with Radial Kernels . . . . . . . . . . . . . . 49


4 Potential and Identity Operators 51

4.1. Riesz Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4.2. Truncated Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.3. One–sided Potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

4.4. Poisson Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60

4.5. Sobolev Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 634.6. Identity Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.7. Partial Sums of Fourier Series . . . . . . . . . . . . . . . . . . . . . . . . 68

v



vi Contents


5 Generalized One-sided Potentials in L p( x) Spaces 71

5.1. Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71

5.2. Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

5.3. Measure of Non–compactness . . . . . . . . . . . . . . . . . . . . . . . . 805.4. Notes and Comments on Chapter 5 . . . . . . . . . . . . . . . . . . . . . . 82

6 Singular Integrals 83

6.1. Hilbert Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2. Cauchy Singular Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

6.3. Riesz Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.4. Calderon–Zygmund Operators . . . . . . . . . . . . . . . . . . . . . . . . 90

6.5. Hilbert Transforms in L p( x) Spaces . . . . . . . . . . . . . . . . . . . . . . 91

6.6. Cauchy Singular Integrals in L p( x) Spaces . . . . . . . . . . . . . . . . . . 98


References 103

Index 119



Preface

One of the important problems of modern harmonic analysis is to establish the bounded-

ness/compactness of integral operators in weighted function spaces. When a given integral

operator is bounded but non-compact, it is natural and useful for applications to have two-

sided sharp estimates of the measure of non-compactness (essential norm) for this operator.

The book is devoted to the boundedness/compactness and weighted estimates of theessential norm for maximal functions, fractional integrals, singular and identity operators,

generally speaking, in weighted variable exponent Lebesgue spaces. Such operators nat-

urally arise in harmonic analysis, boundary value problems for PDE, spectral theory of

differential operators, continuum and quantum mechanics, stochastic processes etc.

One of the main characteristic features of the monograph is that the problems are stud-

ied in the two-weighted setting and cover the case of non-linear maps, such as, Hardy-

Littlewood and fractional maximal functions. Before, these problems were investigated

only for a restricted class of kernel operators consisting only of Hardy-type and Riemann-

Liouville transforms (see, e.g., the monographs [39], [149], [49], [40] and references

therein).

The book may be considered as a systematic and detailed analysis of a class of specific

integral operators from the boundedness/compactness or non-compactness point of view.

There is a wide range of problems of mathematical physics whose solutions are closely

connected to the subject matter of the book. The main subjects of the monograph (maximal

functions, fractional integrals, Hilbert transforms, Riesz transforms, Calderon–Zygmund

singular integrals) are important tools for solving a variety problems in several areas of

mathematics and its applications.

The problems related to estimates of the measure of the non-compactness for differential

and integral operators acting between Banach spaces are closely connected with eigenvalueestimates and other spectral properties for these operators (see, for example, monographs

[39], [64], [17], [40]).

One of the most challenging problems of the spectral theory of differential operators is

the derivation of eigenvalue and singular value estimates for integral operators in terms of

their kernels. The works [13], [197], [135] mark an important stage in the development of

this theory (see also [39], [64], [149], [49]). Until recently the list of non–trivial cases in

which sharp two–sided estimates are available was rather short. Here we present two–sided

estimates of the singular numbers for some classes of kernel operators.

A weight theory for a wide class of integral transforms with positive kernels includingfractional integrals was developed in the monographs [112], [76], [49]. It should be empha-

sized that the interest in fractional calculus has been stimulated by applications in different



viii Alexander Meskhi

fields of science, including stochastic analysis of long memory processes, numerical anal-

ysis, physics, chemistry, engineering, biology, economics and finance. For the theory of

fractional integration and differentiation we refer to the well–known monograph [209].

The book is divided into six chapters and each chapter into sections. The book is

started with some background. We have restricted ourself to the concepts of homogeneous

groups, measure of non–compactness, Schatten–von Neumann ideals, variable exponentLebesgue spaces. Chapter 2 deals with the measure of non–compactness for maximal op-

erators defined, generally speaking, on homogeneous groups. Chapter 3 is focused on the

boundedness/compactness, Schatten–von Neumann ideal norm estimates and measure of

non–compactness for integral operators defined on cones of homogeneous group, while

in Chapter 4 we discuss two-weight estimates of the measure of non–compactness for frac-

tional integral and identity operators defined on Euclidean spaces and homogeneous groups.

In Chapter 5 we establish boundedness/compactness criteria and two–sided estimates of

the measure of non–compactness for the Riemann-Liouville transform in Lebesgue spaces

with non–standard growth. In Chapter 6 we present some results regarding one and twoweighted estimates of the essential norm for singular integrals (Hilbert transforms, Cauchy

integrals, Riesz transforms, Calderon–Zygmund singular integrals), generally speaking in

L p( x) spaces.

One of the important examples of homogeneous groups is the Heisenberg group. Lately

the theory of function spaces on the nilpotent groups has attracted considerable attention

among researchers (see, for example, the monograph by G. Folland and E. Stein [70] ).

This attention has mainly been triggered by questions related to solvability of problems for

differential equations with variable coefficient occurring on manifolds. For example, the

Heisenberg group and function spaces on it have turned out to be closely connected withboundary value problems for pseudo-convex domains in Cn.

The last two chapters of the monograph is dedicated to the investigation of the com-

pactness and non–compactness problems for one–sided potentials and singular integrals,

generally speaking, in weighted Lebesgue spaces with variable exponent.

During the last decade a considerable interest of researchers was attracted to the study of

various mathematical problems in the so called spaces with non–standard growth: variable

exponent Lebesgue and Sobolev spaces L p(·) ,W n, p(·). Such spaces naturally arise when one

deals with functionals of the form

Ω

|∇ f ( x)| p( x)dx.

Such a functional appears, for instance, in the study of differential equations of the type

div (|∇u( x)| p( x)−2∇u) = |u|σ( x)−1u( x) + f ( x).

In this case one deals with the Dirichlet integral of the form Ω

(|∇ f ( x)| p( x) + |u( x)|σ( x))dx.

Such mathematical problems and spaces with variable exponent arise in applications to

mechanics of the continuum medium. In some problems of mechanics there arise varia-

tional problems with Lagrangians more complicated than is usually assumed in variational



Preface ix

calculus, for example, of the form |ξ|γ ( x) when the character of non-linearity varies from

point to point (Lagrangians of the plasticity theory, Langrangians of mechanics of the so

called rheological fluids and others).

Investigation of variational problems with variable exponent started from the papers by

V. Zhikov [238], [239] related to the so called Lavrentiev phenomenon.

M. Ruzicka [202] studied the problems in the so called rheological and electrorheolog-ical fluids, which lead to the spaces with variable exponent.

The variable exponent Lebesgue spaces first appeared in 1931 in the paper by W. Orlicz

[189], where the author established some properties of L p( x) spaces on the real line. Further

development of these spaces was connected with the theory of modular spaces. The first

systematic study of modular spaces is due to H. Nakano [176]. The basis of the variable

exponent Lebesgue and Sobolev spaces were developed by J. Musielak (see [173], [174]),

H. Hudzik [94], I. I. Sharapudinov [220], O. Kovacik and I. Rakosnık [138], S. Samko

[204], [205], etc (see also the surveys [208], [111] and references therein).

The monograph covers not only recent results of the research carried out by the authorand his collaborators regarding the main topics of the monograph, but also contains un-

published material. The monograph includes overview of results of other mathematicians

working on the topics of the book. The bibliography contains 242 titles.

A few words about organization of the book are necessary. The enumeration of theo-

rems, lemmas, formulas etc. follows the natural three-digit system. There are three cat-

egories for numbering: theorems, lemmas, propositions and remarks, and the same for

formulas.

The book is aimed at a rather wide audience, ranging from researchers in functional and

harmonic analysis to experts in applied mathematics and graduate students.I express my gratitude to Professor Vakhtang Kokilashvili for his encouragement to

prepare this monograph, drawing my attention to the problems studied in Chapter 6 and his

remarks and suggestions.

The investigation of the measure of non-compactness of maximal operators started in

2001, when I visited the Centre for Mathematical Analysis and its Applications, University

of Sussex. I expresses my deep gratitude to Professor David Edmunds and the Centre for

support and warm hospitality.

Some aspects of Chapters 2 and 6 were discussed with Professor Alberto Fiorenza. I

am thankful to him for invitation at the University of Naples.

Some results of the monograph were obtained during my stay at the Abdus Salam

School of Mathematical Sciences, GC University, Lahore. I am grateful to Professor A.

D. Raza Choudary for giving me an opportunity to work with PhD students.

Acknowledgement

The monograph was partially supported by INTAS grant No.051000008-8157; No. 06-

1000017-8792, and Georgian National Foundation Grant No. GNSF/ST06/3-010.

Basic Notation

Rn: n-dimensional Euclidean space;

R =R1;

R+ = [0,∞);

C: complex plane;

B( x,r ): open ball with center x and radius r ;¯ B( x,r ): closed ball with center x and radius r ;

I (a,r ) = (a − r ,a + r );

Z : set of all integers;

Z +: set of all non-negative integers;

Z −: set of all non-positive integers;

N: set of all natural numbers;

Bn: volume of the unit ball in Rn, i.e., Bn = 2πn/2

nΓ (n/2) ;

S n−1: area of the unit sphere in Rn, i.e., S n−1 = 2πn/2

Γ (n/2) ;

w( E ) = E w( x)dx, where w is a weight function;| E |: Lebesgue measure of E ;

χ E : characteristic function of a set E ;

p′ = p p−1

, where p is a constant with 1 < p < ∞;

r ′( x) = r ( x)r ( x)−1

, where r is a real-valued function;

limn→∞

an = liminf n→∞

an, limn→∞

an = limsupn→∞

an for a sequence of real number an;

a ≈ b: there are positive constants c1 and c2 such that c1a ≤ b ≤ c2a;

C m: class of functions whose partial derivatives up to and including those of order m exist

and are continuous;

C m0 : subset of C m of functions with compact support;

C 0: class of continuous functions with compact support;

: end of the proof.

Chapter 1

Basic Ingredients

In this chapter definitions and some auxiliary results are given regarding the main objects

of the monograph: homogeneous groups, measure of non–compactness of sublinear andlinear operators, Schatten–von Heumann ideals of compact linear operators, Hardy–type

inequalities, variable exponent Lebesgue spaces and singular integrals.

1.1. Homogeneous Groups

A homogeneous group is a simply connected nilpotent Lie group G on a Lie algebra g

with an one–parameter group of transformations δt = ex p( A log t ), t > 0, where A is a

diagonalized linear operator in G with positive eigenvalues. In the homogeneous group G

the mappings exp o δt o exp−1, t > 0, are automorphisms in G, which will be again denotedby δt . The number Q = tr A is the homogeneous dimension of G. The symbol e will stand

for the neutral element in G.

It is possible to equip G with a homogeneous norm r : G → [ 0,∞) which is continuous

on G, smooth on G\e and satisfies the conditions:

(i) r ( x) = r ( x−1) for every x ∈ G;

(ii) r (δt x) = tr ( x) for every x ∈ G and t > 0;

(iii) r ( x) = 0 if and only if x = e ;

(iv) There exists a constant c0 > 0 such that

r ( xy) ≤ c0(r ( x) + r ( y)), x, y ∈ G.

In the sequel we denote by B(a,ρ) and ¯ B(a,ρ) open and closed balls respectively with

the center a and radius ρ, i.e.

B(a,ρ) := y ∈ G; r (ay−1) < ρ, ¯ B(a,ρ) := y ∈ G; r (ay−1) ≤ ρ.

It can be observed that δρ B(e,1) = B(e,ρ).

Let us fix a Haar measure | · | in G such that | B(e,1)| = 1. Then |δt E | = t Q| E |. In

particular, | B( x,t )| = t Q for x ∈ G, t > 0.

Examples of homogeneous groups are: the Euclidean n-dimensional space Rn, the

Heisenberg group, upper triangular groups, etc. For the definition and basic properties

of the homogeneous group we refer to [70].

2 Alexander Meskhi

Suppose that S is the unit sphere in G, i.e., S = x ∈ G : r ( x) = 1.

Proposition 1.1.1. Let G be a homogeneous group. Then There is a (unique) Radon

measure σ on S such that for all u ∈ L1(G) ,

G

u( x)dx =

∞ 0

S

u(δt y)t Q−1d σ( y)dt .

For the details see, e.g., [70] , p. 14.

Let Ω ⊆ G be a set with positive Haar measure.

Suppose that w be a locally integrable almost everywhere positive function on Ω (i.e. a

weight). Denote by L pw(Ω) (0 < p <∞) the weighted Lebesgue space, which is the space of

all measurable functions f : Ω→ C with the finite norm (quasi-norm if 0 < p < 1)

f L pw(Ω) = Ω | f ( x)| pw( x)dx1/ p

.

If w ≡ 1, then we denote L p1 (Ω) by L p(Ω).

Let A be a measurable subset of S with positive measure. We denote by E a cone in G

defined by

E := x ∈ G : x = δs x, 0 < s < ∞, x ∈ A.

It is clear that if A = S , then E = G.

The next statement is a consequence of Proposition 1.1.1.

Proposition 1.1.2 Let G be a homogeneous group and let A ⊂ S. There is a Radonmeasure σ on S such that for all u ∈ L1( E ) ,

E

u( x)dx =

∞ 0

A

u(δs ¯ y)sQ−1d σ( ¯ y)ds.

Now we formulate embedding criteria from L pw(Ω) to L

qv (Ω) (q < p), where Ω is a non-

empty open set in G. These results are well-known (see [100]) but we give the proofs for

completeness.

Proposition 1.1.3. Let 0 < q < p <∞ and let v and w be weights on an open set Ω⊆ G.

Then L pw(Ω) is boundedly embedded in L

qv (Ω) if and only if

BΩ :=

Ω

v( x)

w( x)

p p−q

w( x)dx

p−q pq

< ∞. (1.1.1)

When BΩ < ∞ , the norm of the embedding I equals BΩ.

Proof. Using Holder’s inequality with respect to the exponent pq

we find that

Ω

| f ( x)|qv( x)dx ≤

Ω

| f ( x)| pw( x)dx

q/ p Ω

v( x)

w( x)

p p−q

w( x)dx

q( p−q) pq

.

Basic Ingredients 3

From this inequality we conclude that

I ≤ BΩ.

To prove the opposite, first we note that from the embedding of L pw(Ω) in L

qv (Ω) it

follows that Ω

v( x)

w( x)

p p−q

w( x)dx < ∞.

Now taking the function f ( x) = v1

p−q ( x)w1

q− p ( x) in the two-weight inequality Ω

| f ( x)|qv( x)dx

1/q

≤ I

Ω

| f ( x)| pw( x)dx

1/ p

we conclude that BΩ < ∞.

The fact that I = BΩ is obvious.

The next statement follows immediately:

Proposition 1.1.4. Let 0 < p <∞. Then L∞(Ω) is boundedly embedded in L pw(Ω) if and

only if

BΩ :=

Ω

w( x)dx

1/ p

< ∞.

When BΩ < ∞ , the norm of the embedding I equals BΩ.

1.2. Measure of Non–compactness

Throughout this section we assume that Ω is either a domain in Rn or a cone in G (of course,

Ω might be G itself).

Let X be a Banach space which is a certain subclass of all measurable functions on Ω.

Denote by X ∗ the space of all bounded linear functionals on X . We say that a real-valued

functional F on X is sub-linear if

(i) F ( f + g) ≤ F ( f ) + F (g) for all non-negative f ,g ∈ X ;

(ii) F (α f ) = |α|F ( f ) for all f ∈ X and α ∈ C.

An operator T : X → L p(Ω) (1 < p < ∞) is said to be sublinear if

T ( f + g)( x) ≤ T ( f )( x) + T (g)( x) almost everywhere

for arbitrary f ,g ∈ X , and

T (α f )( x) = |α|T ( f )( x) almost everywhere

for all non-negative f ∈ X and α ∈C.

Let T be a sublinear operator T : X → L

q

(Ω), where X = L

p

w(Ω). Then the norm of theoperator T is defined as follows:

T = supT f Lq(Ω) : f X ≤ 1.

4 Alexander Meskhi

Moreover, T is order-preserving if T f ( x) ≥ T g( x) almost everywhere for all non-negative f

and g with f ( x) ≥ g( x) almost everywhere. Further, if T is sub-linear and order preserving,

then obviously it is non-negative, i.e. T f ( x) ≥ 0 almost everywhere if f ( x) ≥ 0.

The measure of non-compactness for an operator T which is bounded, order-preserving

and sublinear from X into a Banach space Y will be denoted by T κ ( X ,Y ) ( or simply T κ )

and is defined as

T κ ( X ,Y ) := distT ,κ ( X ,Y ) = inf T − K : K ∈ κ ( X ,Y ),

where κ ( X ,Y ) is the class of all compact sublinear operators from X to Y .

For bounded linear operator T : X → Y , where Y is a Banach space, we denote

T K ( X ,Y ) := distT ,K ( X ,Y ) = inf T − K : K ∈ K ( X ,Y ),

where K ( X ,Y ) is the class of all compact linear operators from X to Y .

If X = Y , then we use the symbol κ ( X ) (resp. K ( X )) for κ ( X ,Y ) (resp. K ( X ,Y )).Let Y be a Banach spaces and let T be a bounded linear operator from X to Y . The

entropy numbers of the operator T are defined as follows:

ek (T ) = inf ε > 0 : T (U X ) ⊂

2k −1 j=1

(bi +εU Y ) for some b1, . . . ,b2k −1 ∈ Y ,

where U X and U Y are the closed unit balls in X and Y respectively.

Let us mention some properties of these numbers (see, e.g., [39]). Suppose that S ,T :

X → Y , R : Y → Z are bounded linear operators, where X ,Y , Z are Banach spaces. Then(i) T = e1(T ) ≥ e2(T ) ≥ · · · ≥ 0;

(ii) em+n−1(S + T ) ≤ em(S ) + en(T ) for all m,n ∈ N;

(iii) em+n−1( RS ) ≤ em( R)en(S ) for all m,n ∈N.

It is known (see, e.g., [64], p. 8) that the measure of non-compactness of T is greater

than or equal to β(T ) := limn→∞

en(T ). Among other properties we mention that β(T ) = 0 if

and only if T ∈ K ( X ,Y ).

We denote by S ( X ) the class of all bounded sublinear functionals defined on X , i.e.,

S ( X ) = F : X →R

, F is sublinear and F = sup x≤1 |F ( x)| < ∞.Let M be the set of all bounded functionals F defined on X with the following property:

0 ≤ F f ≤ Fg

for any f ,g ∈ X with 0 ≤ f ( x) ≤ g( x) almost everywhere. We also need the following

classes of operators acting from X to L p(Ω):

F L( X , L p(Ω)) := T : T f ( x) =m

∑ j=1

α j( f )u j, m ∈ N, u j ≥ 0, u j ∈ L p(Ω),

u j are linearly independent and α j ∈ X ∗

M ,

Basic Ingredients 5

F S ( X , L p(Ω)) :=

T : T f ( x) =m

∑ j=1

β j( f )u j, m ∈N, u j ≥ 0, u j ∈ L p(Ω),

u j are linearly independent and β j ∈ S ( X )

M .

If X = L p(Ω), we denote these classes by F L L p(Ω) and F S L p(Ω) respectively. It is

clear that if P ∈ F S X , L p(Ω) resp. P ∈ F L X , L p(Ω) , then P is compact sublinear resp. compact linear

from X to L p(Ω).

We shall use the symbol α(T ) (resp. α(T )) for the distance between the operator T :

X → L p(Ω) and the class F S

X , L p(Ω)

, (resp. F L

X , L p(G)

) i.e.

α(T ) := dist T ,F S

X , L p(Ω)

( resp. α(T ) := dist T ,F L

X , L p(G)

).

For any bounded subset A of L p(Ω) (1 0 : A can be covered by finitely many open balls in

L p

(Ω) of radius δand

Ψ( A) := inf P∈F L( L p(Ω))

sup

f − P f L p(Ω) : f ∈ A.

We shall need a statement similar to Theorem V.5.1 of [39].

Theorem 1.2.1. Let Ω be a domain in Rn. For any bounded subset K ⊂ L p(Ω) (1 ≤ p < ∞) the inequality

2Φ(K ) ≥Ψ(K ) (1.2.1)

holds.

Proof. Let ε>Φ(K ). Then there exist g1,g2, . . . ,g N from L p(Ω) such that for all f ∈ K

and some i ∈ 1,2, . . . , N ,

f − gi L p(Ω) < ε (1.2.2)

Given δ > 0, let Ω be a cube such that for all i ∈ 1,2, . . . , N Ω\Ω |gi( x)| pdx

1/ p

< 1

2δ. (1.2.3)

We assume that all functions from L p(Ω) are equal to zero outside Ω. Let

Ω = ∪m

j=1Q j,

where the Q j are disjoint congruent cubes of diameter h, and define

P f ( x) :=m

∑ j=1

f Q jχ Q j

( x), f Q j:= | Q j|

−1 Q j

f ( y)dy,

where Q j = Ω∩ Q j. Then

gi − Pgi L p(Ω∩Ω) =m

∑ j=1

Q j

1

| Q j|

Q j

[gi( x) − gi( y)]dy

pdx

≤m

∑ j=1 Q j

1

| Q j| Q j

|gi( x) − gi( y)| pdydx

≤ sup| z|<h

Ω

|gi( x) − gi( x + z)| pdx → 0

6 Alexander Meskhi

as h → 0. From this and (1.2.3) it follows that we may choose h so that

gi − Pgi L p(Ω) < δ, i = 1,2, . . . , N . (1.2.4)

Further,

P f p

L p(Ω) =m

∑ j=1

Q j

| Q j|−1 Q j

f ( y)dy pdx

≤m

∑ j=1

Q j

| Q j|−1

Q j

| f ( y)| pdydx ≤ f p

L p(Ω).

It is obvious that the functionals f → f Q jbelong to ( L p(Ω))∗ ∩M. Consequently, P ∈

F L( L p(Ω)). Finally from (1.2.2)–(1.2.4) and the last inequality we conclude that

f − P f L p(Ω) ≤ f − gi L p(Ω) + gi − Pgi L p(Ω)

+P(gi − f ) L p(Ω) < ε+δ+ gi − f L p(Ω) < 2ε+δ.

Thus we have inequality (1.2.1) because δ is arbitrarily small number.

For the case of homogeneous groups we have the statement similar to the previous one.

Theorem 1.2.2. Let Ω be a cone in G. For any bounded subset K ⊂ L p(Ω) (1 ≤ p <∞)inequality (1.2.1) holds.

Proof. For simplicity assume that E = G. Let ε>Φ(K ). Then there are g1,g2, . . . ,g N ∈

L p(G) such that for all f ∈ K and some i ∈ 1,2, . . . , N

f − gi L p(G) < ε. (1.2.5)

Further, given δ > 0, let ¯ B be the closed ball in G with center e such that for all i ∈1,2, . . . , N

G\ ¯ B

|gi( x)| pdx1/ p

< 1

2 δ. (1.2.6)

It is known (see [70], p. 8) that every closed ball in G is a compact set. Let us

cover ¯ B by open balls with radius h. Since ¯ B is compact, we can choose a finite sub-

cover B1, B2, . . . , Bn. Further, let us assume that E 1, E 2, . . . , E n is a family of pairwise

disjoint sets of positive measure such that ¯ B =n

i=1

E i and E i ⊂ Bi (we can assume that

E 1 = B1 ∩ ¯ B, E 2 = ( B2\ B1) ∩ ¯ B, . . . , E k = ( Bk \k −1i=1

Bi) ∩ ¯ B, . . .). We define

P f ( x) =n

∑i=1

f E iχ E i ( x), f E i = | E i|−1 E i

f ( x)dx.

Basic Ingredients 7

Then

gi − Pgi p

L p( ¯ B) =

n

∑ j=1

E j

1

| E j|

E j

[gi( x) − gi( y)]dy

pdx

≤

m

∑ j=1

E j

1| E j|

E j

gi( x) − gi( y) pdy dx

≤ supr ( z)≤2c0h

¯ B

gi( x) − gi( zx) pdx → 0

as h → 0. The latter fact follows from the continuity of the norm L p(G) (see, e.g., [70], p.

19). From this and (1.2.5) it follows that

gi − Pgi L p(G) < δ, i = 1,2,3, . . . , N , (1.2.7)

when h is sufficiently small. Further,

P f p

L p(G) =n

∑ j=1

E j

| E j|−1 ¯ E j

f ( y)dy

pdx

≤n

∑ j=1

E j

| E j|−1 ¯ E j

f ( y) pdy dx ≤ f p

L p( ¯ B) ≤ f p

L p(G).

It is also clear that the functionals f → f E i belong to L p(G)∗

∩ M . Hence P ∈F L( L p(G)). Finally (1.2.5) − (1.2.7) yield

f − P f L p(G) ≤ f − gi L p(G) + gi − Pgi L p(G) + P(gi − f ) L p(G)

< ε+δ+ gi − f L p(G) ≤ 2ε+δ.

Since δ is arbitrarily small, we have the desired result.

Lemma 1.2.1. Let 1 ≤ p < ∞. Assume that K ⊂ L p(Ω) is compact. Then for any given

ε > 0 there exists an operator Pε ∈ F L L p(Ω) such that for all f ∈ K,

f − Pε f L p(Ω) ≤ ε .

Proof. Suppose that Ω is a cone in G and that K is a compact set in L p(Ω). Using

Theorem 1.2.2 we see that Ψ(K ) = 0. Hence for ε > 0 there exists Pε ∈ F L

L p(Ω)

such

that

sup

f − Pε f L p(Ω) : f ∈ K

≤ ε.

Lemma 1.2.2. Let T : X → L p(G) be compact, order-preserving and sublinear (resp. compact linear ) operator, where 1 ≤ p < ∞. Then α(T ) = 0 (res. α(T ) = 0).

8 Alexander Meskhi

Proof. We prove the lemma for compact and sublinear operators. The proof is the same

in the linear case. For simplicity assume that Ω is a cone in G. Let U X =

f : f X ≤ 1.

From the compactness of T it follows that T (U X ) is relatively compact in L p(Ω). Using

Lemma 1.2.1 we have that for any given ε> 0 there exists an operator Pε ∈ F L

L p(Ω)

such

that for all f ∈ U X ,

T f − PεT f L p(Ω) ≤ ε. (1.2.8)

Let Pε = Pε T . Then Pε ∈ F S

X , L p(Ω)

. Indeed, there exist functionals α j ∈ X ∗ ∩ M , j ∈

1,2, . . . ,m, and linearly independent functions u j ∈ L p(Ω), j ∈ 1,2, . . . ,m, such that

Pε f ( x) = Pε(T f )( x) =m

∑ j=1

α j(T f )u j( x) =m

∑ j=1

β j( f )u j( x),

where β j = α j T belongs to S ( X ) ∩ M . By (1.2.8) we have

T f − Pε f L p(Ω) ≤ ε

for all f ∈ U X , which on the other hand, implies that α(T ) = 0.

We shall also need the following lemma.

Lemma 1.2.3. Let T be a bounded, order-preserving and sublinear operator (resp.

bounded linear operator ) from X to Lq(G) , where 1 ≤ q < ∞. Then

T κ = α(T ) (resp. T K = α(T ))

Proof. Let T be bounded, order preserving and sublinear. Suppose that δ > 0. Thenthere exists an operator K ∈ κ ( X , Lq(Ω)), such that T − K ≤ T κ +δ. By Lemma 1.2.2

there is P ∈ F S ( X , Lq(Ω)) for which the inequality K − P < δ holds. This gives

T − P ≤ T − K + K − P ≤ T κ + 2δ.

Hence α(T ) ≤ T κ . The opposite inequality

T κ ≤ α(T )

is obvious.

Lemma 1.2.4. Let G be a homogeneous group 1 ≤ q < ∞ and let P ∈F S ( X , Lq(G)) (resp. P ∈ F L( X , Lq(G))). Then for every a ∈ G and ε > 0 there exist an

operator R ∈ F S ( X , Lq(G)) (resp. R ∈ F L( X , Lq(G))) and positive numbers α , α such that

for all f ∈ X the inequality

(P − R) f Lq(G) ≤ ε f X

holds and supp R f ⊂ B(a, α)\ B(a,α).

Proof. There exist linearly independent non-negative functions u j ∈ Lq(G), j ∈

1,2, . . . , N , such that

P f ( x) = N

∑ j=1

β j( f )u j( x), f ∈ X ,

Basic Ingredients 9

where β j are bounded, order-preserving, sublinear functionals β j : X → R. On the other

hand, there is a positive constant c for which the inequality

N

∑ j=1

|β j( f )| ≤ c f X

holds.

Let us choose linearly independent Φ j ∈ Lq(G) and positive real numbers α j, α j such

that

u j −Φ j Lq(G) < ε, j ∈ 1,2, . . . , N ,

and suppΦ j ⊂ B(a, α j)\ B(a,α j). If

R f ( x) = N

∑ j=1

β j( f ) Φ j( x),

then it is obvious that R ∈ F S ( X , Lq(G)) and, moreover,

P f − R f Lq(G) ≤ N

∑ j=1

|β j( f )|u j −Φ j Lq(G) ≤ cε f X

for all f ∈ X . Besides this, supp R f ⊂ B(a, α)\ B(a,α), where α = minα j and α =maxα j.

Lemmas 1.2.3 and 1.2.4 for Lebesgue spaces defined on Euclidean spaces have been

proved in [39] in the linear case.

In a similar manner we have the following statements (the proofs are omitted):

Lemma 1.2.5. Let Ω be a domain in Rn and let P ∈ F S

X , L p(Ω)

(resp. P ∈

F L( X , Lq(Ω))) , where X = Lr w(Ω) and 1 < r , p < ∞. Then for every a ∈ Ω and ε > 0 there

exists an operator R ∈ F S

X , L p(Ω)

(resp. P ∈ F L( X , Lq(G))) and positive numbers β1 and

β2 , β1 < β2 such that for all f ∈ X the inequality

(P − R) f

L p(Ω)

≤ ε f X

holds and supp R f ⊂ D(a,β2)\ D(a,β1) , where D(a,s) := Ω B(a,s).

Lemma 1.2.6. Let Ω be a cone in G. Suppose that 1 ≤ p <∞ and Y = L pv (Ω). Suppose

that P ∈ F L( X ,Y ) and ε > 0. Then there are an operator T ∈ F L( X ,Y ) and a set E α,β := x ∈Ω : 0 < α < r ( x) < β < ∞ such that

P − T < ε

and

supp T f ⊂ E α,β

for every f ∈ X.

Lemma 1.2.7. Let G be a homogeneous group. Suppose that 1 < p,q < ∞ and T a

bounded, order-preserving and sublinear (resp. bounded liear ) operator from L pw(G) to

10 Alexander Meskhi

Lqv (G). Suppose that λ > T κ ( L p

w(G), Lqv (G)) (resp. λ > T K ( L

pw(G), L

qv (G))) and a is a point of

G. Then there exist constants β1,β2, 0 < β1 < β2 <∞, such that for all τ and r with r > β2 ,τ < β1, the following inequalities hold:

T f Lqv ( B(a,τ)) ≤ λ f L

pw(G), (1.2.9)

T f Lqv ( B(a,r )c) ≤ λ f L

pw(G), (1.2.10)

where f ∈ L pw(G).

Proof. Let T be bounded, order preserving and sublinear from L pw(G) to L

qv (G). Let

T (v) be the operator given by

T (v) f = v1q T f .

Then it is easy to see that

T (v)κ ( L pw(G)→ Lq(G)) = T κ ( L p

w(G)→ Lqv (G)).

By Lemma 1.2.3 we have that

λ > α

T (v).

Consequently, there exists P ∈ F S

L

pw(G), Lq(G)

such that

T (v) − P < λ.

Fix a ∈ G. According to Lemma 1.2.4 there are positive constants β1 and β2, β1 < β2, and

R ∈ F S L pw(G), L

qv (G) for which

P − R ≤ λ− T (v) − P

2

and supp R f ⊂ B(a,β2)\ B(a,β1) for all f ∈ L pw(G). Hence,

T (v) − R < λ.

From the last inequality it follows that if 0 < τ< β1 and r > β2, then (1.2.9) and (1.2.10)

hold for f ∈ L

p

w(G).

The next statement follows in a similar manner as Lemma 1.2.7; therefore the proof is

omitted.

Lemma 1.2.8. Ω be a domain in Rn. Suppose that 1 < p,q <∞ and that T is bounded,

order-preserving and sublinear (resp. bounded linear ) operator from L pw(Ω) to L

qv (Ω).

Assume that λ > T κ ( L pw(Ω), Lq

v (Ω)) (resp. λ > T K ( L pw(Ω), Lq

v (Ω))) and a ∈ Ω. Then there

exist constants β1,β2,0 < β1 < β2 < ∞ such that for all τ and r with r > β2 , τ < β1 , the

following inequalities hold:

T f Lqv (Ω∩ B(a,τ)) ≤ λ f L pw(Ω); T f Lqv (Ω\ B(a,r )) ≤ λ f L pw(Ω),

where f ∈ L pw(Ω).

Basic Ingredients 11

1.3. Hardy–type Transforms

In this section we give some Hardy-type inequalities.

Theorem 1.3.1 ([49] (Section 1.1)). Let ( X , µ) be a measure space and let φ : X → R

be a µ-measureable non-negative function. Suppose that 1 0 x:φ( x)>t

v( x)dµ1/q

x:φ( x)<t

w1− p′( x)dµ

1/ p′

< ∞, p′ = p/( p − 1).

Let E be a cone in G, where G is a homogeneous group (see Section 1.1).

Denote

E t := y ∈ E : r ( y) < t ,

where t is a positive number.

Taking X = E , dµ = dx, v = u, w = 1, φ( x) = r ( x) in the previous statement and observ-

ing that

supt >0 E \ E t u( x)dx1/q

t

Q/ p′

≈ sup j∈Z E 2 j+1 \ E

2 j u( x)dx1/q

2

jQ/ p′

,

we have the next statement.

Theorem 1.3.2. Let 1 < p ≤ q < ∞. Then the inequality E

v( x)

E r ( x)

f ( y)dy

qdx

1/q

≤ c

E

| f ( x)| pdx

1/ p

holds if and only if

sup j∈Z

E 2 j+1 \ E

2 j

u( x)dx1/q

2 jQ/ p′< ∞.

We need also the following statement (see [158] for 1 ≤ q 1. Then the inequality ∞0

v( x)

x

0 f (t )dt

q

dx

1/q

≤ c

∞0

( f ( x)) pw( x)dx

1/ p

, f ≥ 0,

holds if and only if ∞0

∞t

v( x)dx

t

0w1− p′

( x)dx

q−1 p/( p−q)

w1− p′(t )dt

( p−q)/( pq)

< ∞.

12 Alexander Meskhi

The next statement deals with the Hardy-inequality on an interval.

Theorem 1.3.4. Let r and s be constants such that 1 < r ≤ s <∞. Suppose that 0 ≤ a <b ≤∞. Let v and w be non-negative measurable functions on [a,b). Then the inequality

b

av( x) x

a f (t )dt s

dx1/s

≤ c b

a(w(t ) f (t ))r dt 1/r

, f ≥ 0,


supa≤t ≤b

b

t vs( x)dx

1/s t

aw−r ′ ( x)dx

1/r ′

< ∞.

The next statements will be useful for us.

Theorem 1.3.5 ([101] (Ch. XI)). Let ( X , µ) and (Y , ν) be σ-finite measure spaces and let 1 < p,q < ∞. Suppose that for positive function a : X ×Y → R , we havea( x, y) L

p′ ν (Y )

L

q µ( X )

< ∞.

Then the operator

A f ( x) =

Y

a( x, y) f ( y)d ν( y)

is compact from L p ν(Y ) to L

q µ( X ).

The next lemma is known as Ando’s theorem (see [4] and [139], Sections 5.3 and 5.4)which in our case is formulated for the set E .

Theorem 1.3.6. Let 0 < q < ∞ , 1 < p < ∞ and q < p. Suppose that v and w are

Haar-measurable almost everywhere positive functions on E. If the operator

A E f ( x) =

E a( x, y) f ( y)dy, x ∈ E ,

is bounded from L pw( E ) to L

qv ( E ) , then A E is compact.

1.4. L p( x) Spaces

Let Ω be a domain in Rn and let p be a measurable function on Ω. Throughout this section

we assume that 1 < p− ≤ p( x) ≤ p+ < ∞, where p− and p+ are respectively the infimum

and the supremum of p on Ω. Suppose that ρ is a weight function on Ω, i.e. ρ is an almost

everywhere positive locally integrable function on Ω. We say that a measurable function f

on Ω belongs to L p(·)ρ (Ω) (or L

p( x)ρ (Ω)) if

S p,ρ( f ) = Ω f ( x)ρ( x) p( x)dx < ∞.

It is known (see, e.g., [138], [111]) that L p( x)ρ (Ω) is a Banach space with the norm

f L

p( x)ρ (Ω)

= inf λ > 0 : S p,ρ

f /λ

≤ 1

.

If ρ ≡ 1, then we use the symbol L p( x)(Ω) (resp. S p) instead of L p( x)ρ (Ω) (resp. S p,ρ). It

is clear that f L

p(·)

ρ (Ω)

= f ρ L p(·)(Ω).

Further, we denote

p−( E ) = inf E

p, p+( E ) = sup E

p

for a measurable set E ⊆ Ω.

Let P (Ω) be the class of all measurable functions p, p : Ω→ Rn, such that the Hardy-

Littlewood maximal operator

M Ω f ( x) = sup

Q∋ x

1

|Q| Q∩Ω

| f ( y)|dy, x ∈Ω,

where the supremum is taken over all cubes Q containing x and satisfying |Q ∩Ω| > 0, is

bounded in L p(·)(Ω).

Throughout the paper we will assume that I (a,r ) is the interval (a − r ,a + r ).

The following lemma is well-known (see e.g., [138], [204]):

Lemma 1.4.1. Let E be a measurable subset of Ω. Then the following inequalities hold:

f p+( E )

L p(·)( E ) ≤ S p( f χ E ) ≤ f

p−( E )

L p(·)( E ), f L p(·)( E ) ≤ 1;

f p−( E )

L p(·)( E ) ≤ S p( f χ E ) ≤ f

p+( E )

L p(·)( E ), f L p(·)( E ) ≥ 1;

E

f ( x)g( x)dx

≤ 1

p−( E ) +

1

( p+( E ))′

f L p(·)( E ) g

L p′(·)( E ),

where p′( x) = p( x) p( x)−1

and 1 < p−( E ) ≤ p( x) ≤ p+( E ) < ∞.

Definition 1.4.1. Let Ω be a domain in Rn. We say that p satisfies weak Lipschitz

(log − Holder continuity) condition ( p ∈ W L(Ω)) if there is a positive constant A such that

for all x and y in Ω with 0 < | x − y| < 1/2 the inequality

| p( x) − p( y)| ≤ A/(− ln | x − y|) (1.4.1)

holds.

The next result is due to L. Diening [29].

Theorem 1.4.1. Let Ω be bounded domain in Rn. If p satisfies the condition (1.4.1) ,then the operator M Ω is bounded in L p(·)(Ω).

Theorem 1.4.2 ([27], [22]). Let Ω = Rn. Then M is bounded in L p(·)(Rn) if (1.4.1)

holds for all x, y ∈Rn with 0 < | x − y| < 1/2 and, moreover, there exists a positive constant

b such that

| p( x) − p( y)| ≤ b/ ln(e + | x|), x, y ∈Rn, | y| ≥ | x|. (1.4.2)

14 Alexander Meskhi

Let

H f ( x) = limε→0

| x−t |>ε

f (t )

x − t dt

be the Hilbert transform.

Theorem 1.4.3 ([26]). Let p ∈ P (R). Then H is bounded in L p(·)(R).

The next statement is also valid (see [59], [206], [124], [91]).

Theorem 1.4.4. The class C 0(Ω) of continuous compactly supported functions on Ω is

dense in L p(·)(Ω).

For other properties of L p( x) and L p( x)ρ spaces see e.g. [138], [220], [204], [111], [208].

We need the following slight modification of Lemma 2.1 from [29] (see also [27]).

Proposition 1.4.1. Let Ω be an open set in R. Suppose that p satisfies condition (1.4.1)

on Ω with the constant A. Then for all intervals I with | I ∩Ω| > 0 and | I | < 1/4 theinequality

| I | p−( I ∩Ω)− p+( I ∩Ω) ≤ e A

holds.

Proof. Let I := (a − r ,a + r ) for some a ∈R and r > 0. It is easy to see that

(2r ) p−( I ∩Ω)− p+( I ∩Ω) ≤ (2r ) A/ ln(2r ) = e A.

Proposition 1.4.2 ([203, 138]). Let |Ω| < ∞ and 1 ≤ r ( x) ≤ p( x) ≤ p < ∞ for x ∈ Ω.

Then L p( x)

(Ω) ⊆ Lr ( x)

(Ω) and

f Lr ( x)(Ω) ≤ (1 + |Ω|) f L p( x)(Ω).

Remark 1.4.1. If p satisfies (1.4.1) with the constant A, then the function 1/ p satisfies

the same condition with the constant A/( p−)2. Indeed, we have

|1/ p( x) − 1/ p( y)| = | p( x) − p( y)|/| p( x) p( y)|

≤ − A1/ ln | x − y|,

where A1 = A/( p−)2 and | x − y| < 1/2. Analogously we can show that if p satisfies (1.4.1)with the constant A, then p′ satisfies the same condition with the constant A/( p− − 1)2.

Let X be a Banach space and let

F L( X , L p(·)(Ω)) ≡

T : T f ( x) =m

∑ j=1

β j( f )u j, m ∈N, u j ≥ 0, u j ∈ L p(·)(Ω),

where u j are linearly independent and β j are bounded linear functionals on L p(·)(Ω). If

X = L p(·)(Ω), then we denote F L( X , L p(·)(Ω)) := F L( L p(·)(Ω)).

For any bounded subset A of L p(·)(Ω), let

Φ( A) := inf δ > 0 : A can be covered by finite open balls in L p(·)(Ω)

of radius δ

and

Ψ( A) := inf P∈F L( L p(·)(Ω))

sup f − P f L p(·)(Ω) : f ∈ A.

The following lemma is similar to Theorem 1.2.1.

Lemma 1.4.2. Let p ∈ P (Ω). Then there exists a positive constant B such that for anybounded subset K ⊂ L p(·)(Ω) the inequality

BΦ(K ) ≥Ψ(K )

holds.

Proof. Let ε > Φ(K ). Then there exist g1, . . . ,g N from L p(·)(Ω) such that for all f ∈ K

and some j ∈ 1, . . . , N ,

f − g j L p(·)(Ω) < ε.

Let us take δ > 0. Then due to Theorem 1.4.4 there are g j ∈ C 0(Ω), such that for all

j ∈ 1, . . . , N ,

g j − g j L p(·)(Ω) < δ. (1.4.3)

Hence for that given δ, f − g j L p(·)(Ω) < ε+δ. (1.4.4)

By the absolutely continuity of the norm there is a cube Q such that

gi L p(·)(Ω\ Q) < δ. (1.4.5)

Suppose that σ is a small positive number. Let us divide Q by disjoint cubes Qi so that

diam Qi < h and h is sufficiently small for which

|g j( x) − g j( y)| < σ

when | x − y| < h, x, y ∈ Rn (in fact, we may assume that g j are extended by 0 continuously

on Rn).

Further, we take

Pφ( x) =m

∑i=1

φQiχ Qi( x); φQi = |Qi|−1

Qi

φ( y)χΩ( y)dy,

where Qi = Qi ∩Ω. Taking now into account the fact that g jχΩ = g j, we have

g j − Pg j p+(Ω∩ Q)

L p(·)(Ω∩ Q) ≤∑

i

Qi

1

|Q|i

Qi

|(g j( x) − gi( y)|dy

p( x)

dx

≤ σ p−(Ω∩ Q)m

∑i=1

| Qi| ≤ σ p− | Q|.

Consequently,

g j − Pg j L p(·)(Ω∩ Q) ≤ | Q|1/ p+σ p−/ p+.

16 Alexander Meskhi

Further, the condition p ∈ P (Ω) yields

Pφ L p(·)(Ω) ≤

N

∑i=1

φQ jχ Q j

L p(·)(Ω)

≤ M Ωφ L p(·)(Ω) ≤ M Ωφ L p(·)(Ω).

Finally, taking into account estimates (1.4.3)–(1.4.5) and the fact that P ∈F L( L p(·)(Ω)), we find that

f − P f L p(·)(Ω) ≤ f − g j L p(·)(Ω) + g j − Pg j L p(·)(Ω) + Pg j − P f L p(·)(Ω)

< ε+ 2δ+ | Q|1/ p+σ p−/ p+ + M Ωg j − f L p(·)(Ω)

≤ (1 + M Ω)ε+ (2 + M Ω)δ+ | Q|1/ p+σ p−/ p+ .

As δ and σ are arbitrarily small and σ is independent of Q, we have the desired result.

Lemma 1.4.3. Let p ∈ P (Ω) and let K ⊂ L p(·)

(Ω) be compact. Then for a given ε > 0there exists an operator Pε ∈ F L( L p(·)(Ω)) such that for all f ∈ K,

f − Pε f L p(·)(Ω) ≤ ε.

Proof. Let K be a compact subset of L p(·)(Ω). By Lemma 1.4.2 we have that Ψ(K ) = 0.Hence, for ε > 0 there exists Pε ∈ F L( L p(·)(Ω)) such that

sup f − Pε f L p(·)(Ω) : f ∈ K ≤ ε.

Let X and Y be Banach spaces. As before (see Section 1.2 for classical Lebesgue

spaces), we denote

α(T ) := distT ,F L( X ,Y ).

Lemma 1.4.4. Let p ∈ P (Ω) and let X be a Banach space. Suppose that T : X → L p(·)(Ω) is a compact linear operator. Then

α(T ) = 0.

Proof. Let U X := f : f X ≤ 1. From the compactness of T it follows that T (U X ) is

relatively compact in L p(·)(Ω). By Lemma 1.4.3 we have that for any ε > 0 there exists an

operator Pε ∈ F L( L p(·)(Ω)) such that for all f ∈ U X ,

T f − PεT f L p(·)(Ω) ≤ ε. (1.4.6)

Let Pε = Pε T . Then Pε ∈ F L( X , L p(·)(Ω)). Indeed, there are bounded linear functionals

α j and linearly independent functions u j ∈ L p(·)(Ω), j ∈ 1, . . . ,m, such that

Pε f ( x) = Pε(T f )( x) =m

∑ j=1

α j(T f )u j( x) =m

∑ j=1

β j( f )u j( x),

where β j = α j T (1 ≤ i ≤ m) are bounded linear functional from X to L p(·)(Ω). Further,

inequality (1.4.6) impliesT f − Pε f L p(·)(Ω) ≤ ε

for all f ∈ U X , from which it follows that α(T ) = 0.

Lemma 1.4.5. Let P ∈ F L( X , L p(·)(Ω)) , where X is a Banach space. Then for every

a ∈Ω and ε > 0 there exist an operator R ∈ F L( X , L p(·)(Ω)) and positive numbers α and α ,

α < α , such that for all f ∈ X the inequality

(P − R) f L p(·)(Ω) ≤ ε f X

holds, and moreover, supp R f ⊂ D(a, α) \ Q(a,α) , where Q(a,r ) is a cube with center a and

side length r; D(a,r ) := Q(a,r ) ∩Ω.

Proof. Since P ∈ F L( X , L p(·)(Ω)), there exist linearly independent non-negative func-

tions u j ∈ L p(·)(Ω), j ∈ 1, . . . ,m, such that

P f ( x) =m

∑ j=1

β j( f )u j( x), f ∈ X ,

where β j are bounded linear functionals on X . Further, there is a positive constant c forwhich N

∑ j=1

|β j( f )| ≤ c f X .

Let us choose linearly independent Φ j ∈ L p(·)(Ω) and positive real numbers α j, α j,

such that

u j −Φ j L p(·)(Ω) < ε, j ∈ 1, . . . , N ,

and suppΦ j ⊂ D(a, α j) \ Q(a,α j).

Let us take

R f ( x) = N

∑ j=1

β j( f )Φ j( x).

Then it is obvious that R ∈ F L( X , L p(·)(Ω)) and, also

P f − R f L p(·)(Ω) ≤ N

∑ j=1

|β j( f )|u j −Φ j L p(·)(Ω) ≤ cε f X

for all f ∈ X . Moreover,

supp R f ⊂

D(

a, α

) \Q

(a,α

),where α = minα j, α = maxα j.

The next lemma will be useful.

Lemma 1.4.6. If f /∈ L p(·)(Ω) , then there exists a non-negative g ∈ L p′(·)(Ω) such that

f g /∈ L1(Ω).

Proof. The easier way to get this result is not a generalization of the arguments from

[88]. In our case we use Landau’s resonance theorem (see e.g. Lemma 2.6, p. 10 of

[12]) according to which a measurable function f belong to the dual space X ′ of a Banach

function space X if and only if f g is integrable for every g in X . In order to use this result

in our case it is enough to note that the dual space of L p(·)(Ω) is L p′(·)(Ω) (see, e.g., [138],

[204]).

18 Alexander Meskhi

Lemma 1.4.7([204]). For all f ∈ L p(·)(Ω) the inequality

f L p(·)(Ω) ≤ supg

L p′(·)(Ω)≤1

Ω

f ( y)g( y)dy

holds.

In the sequel we will assume that

E k := [2k ,2k +1); I k := [2k −1,2k +1), k ∈ Z.

The next statements will be useful in Chapter 5.

Lemma 1.4.8. Let 1 < p−(R+) ≤ p( x) ≤ q( x) ≤ q+(R+) < ∞ and let p,q ∈ W L(R+).

Suppose that p( x) ≡ pc = const , q( x) ≡ qc = const when x > a for some positive number a.

Then there exists a positive constant c such that

∑k

f χ I k L p(·)(R+)gχ I k

Lq′(·)(R+) ≤ c f L p(·)(R+)g Lq′(·)(R+)

for all f ∈ L p(·)(R+) and g ∈ Lq′(·)(R+).

Proof. For simplicity assume that a = 1. Let us split the sum as follows:

∑i

f χ I i L p(·)(R+)gχ I i Lq′(·)(R+) =∑i≤2

+∑i>2

:= J 1 + J 2.

Taking into account that p( x) = pc = const, q( x) ≡ qc ≡ const on the set (1,∞), using

Holder’s inequality for series and the fact that pc ≤ qc, we have

J 2 =∑i>2

f χ I i L pc (R+)gχ I i L(qc)′(R+) ≤ c f L p(·)(R+)g

Lq′(·)(R+).

Now let us estimate J 1. Suppose that f L p(·)(R+) ≤ 1 and g Lq′(·)(R+) ≤ 1. First notice that

q,q′ ∈ W L(R+). Therefore, by Lemma 1.4.1 and Proposition 1.4.1 we have

| I k |1/q+( I k ) ≈ χ I k

Lq(·)(R+) ≈ | I k |1/q−( I k );

| I k |1/(q′)+( E k ) ≈ χ E k

Lq′(·)(R+) ≈ | I k |

1/(q′)−( I k ),

where k ≤ 2. Hence Holder’s inequality (see Lemma 1.4.1) yields

J 1 ≤ c∑k ≤2

8

0


Lq′(·)(R+)

χ I k Lq(·)(R+)χ E k

Lq′(·)(R+)

χ E k ( x)dx

≤ c

8

0∑k ≤2


Lq′(·)(R+)

χ I k Lq(·)(R+)χ I k

Lq′(·)(R+)

χ E k ( x)dx

≤ c∑k ≤2

f χ I k L p(·)(R+)

χ I k Lq(·)(R+)

χ E k (·)

Lq(·)((0,8))

∑k ≤2

gχ I k L p′(·)(R+)

χ I k

L p′(·)(R+)

χ E k (·)

Lq′(·)((0,8))

:= cS 1 · S 2.




Now we claim that

I (q) ≤ cI ( p),

where

I (q) := ∑k ≤2

f χ I k L p(·)(R+)

χ I k Lq(·)(R+) χ

E k (·) Lq(·)((0,8))

;

I ( p) :=

∑k ≤2

f χ I k L p(·)(R+)

χ I k L p(·)(R+)

χ E k (·)

L p(·)((0,8))

.

Indeed, suppose that I ( p) ≤ 1. Further, Proposition 1.4.1 and Lemma 1.4.1 yield

∑k ≤2

1

| I k |

E k

f χ I k

p( x)

L p(·)(R+)dx ≤ c

8 0 ∑k ≤2

f χ I k L p(·)(R+)

χ I k L p(·)(R+)

χ E k ( x)

p( x)

dx ≤ c.

Consequently, taking into account that q( x) ≥ p( x), E k ⊂ I k and f L p(·)(R+) ≤ 1, we find

that

∑k ≤2

1

| I k |

E k

f χ I k

q( x)

L p(·)(R+)dx ≤ c∑

k ≤2

1

| I k |

E k

f χ I k

p( x)

L p(·)(R+)dx ≤ c.

This implies that I (q) ≤ c.

Let us introduce a function

P(t ) = ∑k ≤2

p+( I k )χ E k (t ).

It is clear that p(t ) ≤ P(t ) because E k ⊂ I k . Hence, Proposition 1.4.2 for Ω = (0,8) yields

I ( p) ≤ c

∑k ≤2

f χ I k L p(·)(R+)

χ I k L p(·)(R+)

χ E k (·)

LP(·)((0,8))

.

Then, using the definition of P and the inequality χ I k

p+( I k )

L p(·)(R+) ≥ c2k , we have

8

0∑

k ≤2

f χ I k L p(·)(R+)χ I k

L p(·)(R+)

χ E k ( x)P( x)

dx

=

8

0

∑k ≤2

f χ I k

p+( I k )

L p(·)(R+)

χ I k

p+( I k )

L p(·)(R+)

χ E k ( x)

dx

≤ c

8

0

∑k ≤2

f χ I k

p+( I k )

L p(·)(R+)

2k χ E k

( x)

dx

≤ c∑k ≤2 f χ I k

p+( I k )

L p(·)(R+) ≤ c∑k ≤2 I k | f ( x)|

p( x)

dx

≤ c

R+

| f ( x)| p( x)dx ≤ c.

20 Alexander Meskhi

Consequently, the estimates derived above give us

S 1 ≤ c f L p(·)(R+).

Analogously, taking into account the fact that q′ ∈ W L(R+) and arguing as above, we

find thatS 2 ≤ cg

Lq′(·)(R+).

Notice that Lemma 1.4.8 is a slight modification of Theorem 2 from [136] (see also [53]

for the case p( x) = q( x)).

Lemma 1.4.9. Let I = [0,a] be a bounded interval and let p ∈ W L( I ). Suppose that

1 < p−( I ) ≤ p+( I ) < ∞ and α( x) > 1/ p( x) when x ∈ I. Then

I ( x) := ( x − ·)α( x)−1χ( x/2, x)(·) L p′(·)(R+) ≤ cxα( x)−1/ p( x),

where the positive constant c does not depend on x.

Proof. First notice that the condition p ∈ W L( I ) implies p′ ∈ W L( I ). Hence we have

the following two-sided estimate:

( x − t ) p′(t ) ≤ c1( x − t ) p′( x) ≤ c2( x − t ) p′(t ),

where 0 < t < x < a and the positive constants c1 and c2 depend only on p and a. Conse-

quently,

x

x/2( x − t )(α( x)−1) p′(t )dt ≤ c x

x/2( x − t )(α( x)−1) p′( x)dt

= c

x/2

0u(α( x)−1) p′( x)du = c( x/2)(α( x)−1) p′( x)+1

= cx(α( x)−1) p′( x)+1 := S ( x).

Suppose that I ( x) ≤ 1. By the fact that 1/ p ∈ W L( I ), Proposition 1.4.1 and Lemma 1.4.1

we have

I ( x) ≤ (S ( x))1/( p′)

+([ x/2, x])

= c( x/2)1/( p′)

+([ x/2, x])(α( x)−1) p′( x)+1

≤ c

( x/2)1/( p′)−([ x/2, x])(α( x)−1) p′( x)+1

≤ c

( x/2)1/ p′( x)(α( x)−1) p′( x)+1

≤ cxα( x)−1/ p( x).

For I ( x) > 1, the conclusion is trivial.

Now we formulate some Hardy-type inequalities in L p(·) spaces.

Let

H f (t ) = x

0 f (t )dt , x > 0,

be the Hardy transform.

Theorem 1.4.5 ([136]). Let p ∈ W L( I ) , where I = [0,a] , where 0 < a < ∞. Suppose

that 1 < p−( I ) ≤ p( x) ≤ q( x) ≤ q+( I ) < ∞ and p,q ∈ W L( I ). Then the Hardy operator H

is bounded from L p(·)w ( I ) to L

q(·)v ( I ) if and only if

C := sup

0<t <a

v(·)χ(t ,a)(·) Lq(·)( I )w−1(·)χ(0,t )(·) L p′(·)( I ) < ∞.

Moreover, there exist positive constants c1 and c2 such that c1C ≤ H ≤ c2C.

To formulate the next results we need the notation:

p0( x) := inf y∈[0, x]

p( y); p0( x) :=

p0( x), 0 ≤ x ≤ a

pc ≡ const, x < a,

where a is a fixed positive number.

Theorem 1.4.6. Let I = [0,a] (0 < a <∞) and let 1 ≤ p−( I ) ≤ p0( x) ≤ q( x) ≤ q+( I ) <∞ for almost every x ∈ I. Then the condition

sup0<t <a

a

t (v( x))q( x)t q( x)/( p0)′( x)dx < ∞

implies the boundedness of H from L p(·)( I ) to Lq(·)v ( I ).

Theorem 1.4.7. Let I = R+ and let 1 ≤ p−( I ) ≤ p0( x) ≤ q( x) ≤ q+( I ) < ∞ for almost

every x ∈ I. Suppose that q( x) ≡ qc = const , p( x) ≡ pc = const when x > a for some positive

number a. Then the condition

sup0<t <∞

∞t

(v( x))q( x)t q( x)/( p0)′( x)dx < ∞

guarantees the boundedness of H from L p(·)( I ) to Lq(·)v ( I ).

Theorems 1.4.6 and 1.4.7 are special cases of Theorems 3.1 and 3.3 of [52] respectively.

Theorem 1.4.8 ([57]). Let p( x) and q( x) be measurable functions on an interval I ⊆ R+.

Suppose that 1 < p−( I ) ≤ p+( I ) < ∞ and 1 < q−( I ) ≤ q+( I ) < ∞. If k ( x, y) L p′( y)( I )

Lq( x)( I )

< ∞,

where k is a non-negative kernel, then the operator

K f ( x) = I

k ( x, y) f ( y)dy

is compact from L p(·)( I ) to Lq(·)( I ).

22 Alexander Meskhi

1.5. Schatten–von Neumann Ideals

Let H be a separable Hilbert space and let σ∞( H ) be the class of all compact linear operators

T : H → H , which form an ideal in the normed algebra B of all bounded linear operators

on H . To construct a Schatten-von Neumann ideal σ p( H ) (0 < p ≤ ∞) in σ∞( H ), the

sequence of singular numbers s j(T ) ≡ λ j(|T |) is used, where the eigenvaluesλ j(|T |) (|T | ≡(T ∗T )1/2) are non-negative and are repeated according to their multiplicities and arranged

in decreasing order. A Schatten-von Neumann quasinorm (norm if 1 ≤ p ≤∞) is defined as

follows:

T σ p( H ) ≡∑

j

s p j (T )

1/ p

, 0 < p < ∞,

with the usual modification if p = ∞. Thus we have T σ∞( H ) = T and T σ2( H ) is the

Hilbert-Schmidt norm given by the formula

T σ2( H ) = |a( x, y)|2

dxdy1/2

for the integral operator

A f ( x) =

a( x, y) f ( y)dy.

We refer, for example, to [13], [14], [16], [135], for more information concerning

Schatten-von Neumann ideals.

Let ρ be a weight function on E , where E is a cone in a homogeneous group G (see the

previous section).

Suppose that

E t := y ∈ E : r ( y) < t ,where t is a positive number.

We denote by l p L2ρ( E )

the set of all measurable functions g : E → R for which

gl p

L2ρ( E ) =

∑

n∈Z

E

2n+1 \ E 2n

|g( x)|2ρ( x)dx p/21/ p

< ∞.

The next statement deals with interpolation result which is a consequence of more gen-

eral statements from [229]( p.127, p. 147) (see also [11]).

Proposition 1.5.1. Let 1 ≤ p0, p1 ≤ ∞, 0 ≤ θ ≤ 1, 1 p = 1−θ p0 + θ p1 . Suppose that ρ is aweight function on E. If A is a bounded operator from l pi

L2ρ( E )

into σ pi

L2ρ( E )

, where

i = 0,1 , then it is also bounded from l p L2ρ( E )

to σ p

L2ρ( E )

. Moreover,

Al p( L2ρ( E ))→σ p( L2( E )) ≤ T 1−θ

l p0 ( L2ρ( E ))→σ p0

( L2( E ))T θ

l p1 ( L2ρ( E ))→σ p1

( L2( E )).

The following statement is obvious for p =∞; when 1 ≤ p <∞ it follows from Lemma

2.11.12 of [197].

Proposition 1.5.2. Let 1 ≤ p ≤ ∞ and let f k , gk be orthonormal systems in a

Hilbert space H . If T ∈ σ p( H ) , then

T σ p( H ) ≥∑

n

|T f n,gn| p1/ p

.

1.6. Singular Integrals in Weighted Lebesgue Spaces

In this section we give some well-known statements regarding singular integrals in weighted

Lebesgue spaces. Let

H f ( x) = p.v. R f ( y)

x − ydy

and

S Γ f (t ) = p.v. 1

πi

Γ

f (τ)

τ− t d τ

be the Hilbert transform and Cauchy singular integral operator respectively, where Γ = t ∈C : t = t (s),0 ≤ s ≤ l ≤ ∞ is a smooth Jordan Curve on which the arc–length is chosen as

a parameter and l is the length of Γ .

Definition 1.6.1. We say that the weight w ∈ A p(R) , 1 0

A(r ,a) p (R) := sup

a∈Rr >0

1

2r

a+r

a−r w(s)ds1/ p 1

2r

a+r

a−r w1− p′

(s)ds1/ p′

< ∞.

Theorem 1.6.1 ([95]). Let 1 < p < ∞. Then the Hilbert transform H is bounded in

L pw(R) if and only if w ∈ A p(R).

The next statement is from [109].

Theorem 1.6.2. Let 1 < p < ∞ , l ≤ ∞. Suppose that Γ is a smooth curve. Then the

inequality l

0|S Γ f (t (s))| pw(s)ds ≤ c

l

0| f (t (s))| pw(s)ds

holds if and only if w ∈ A p(0, l) , that is,

Bl := sup

1

| I |

I

w(s)ds

1/ p 1

| I |

I

w1− p′(s)ds

1/ p′

< ∞,

where the supremum is taken over all intervals in (0, l).

For the next statement see, e.g., [172], [46], [49], p. 553.

Proposition 1.6.1. Let 1 0

| x|>t

v( x)

| x| p dx

1/ p | x|<t

w1− p′( x)dx

1/ p′

< ∞

and

supt >0

| x|<t

v( x)dx

1/ p | x|>t

w1− p′( x)

| x| p′ dx

1/ p′

< ∞.

Suppose that for the operator

K f ( x) = p.v.

Rn

k ( x − y) f ( y),dy (1.6.1)

24 Alexander Meskhi

the inequality K f

L2(Rn) ≤ c

f

L2(Rn), f ∈ C ∞0 (Rn) (1.6.2)

holds and the kernel k satisfies the following two conditions:

(i) there exists a positive constant A such that the inequality

∂∂ x

αk ( x)

≤ A| x|−n−α (1.6.3)

holds for all x ∈Rn, x = 0, and |α| ≤ 1;

(ii) there exists a positive constant b and an unit vector u0 such thatk ( x)≥ b| x|−n (1.6.4)

when x = λ · u0 with −∞ < λ < +∞.It is easy to see that the Riesz transforms

R j f ( x) = limr →0γ n

Rn\ B( x,r )

x j − y j

| x − y|n+1 f ( y) dy, j = 1, . . . ,n, (1.6.5)

where x = ( x1, . . . , xn) ∈ Rn, γ n = Γ [(n + 1)/2]/π(n+1)/2, satisfy conditions (1.6.2)–(1.6.4).

If n = 1, then R1 f ( x) is the Hilbert transform H .

Definition 1.6.2. Let 1 < p < ∞. We say that the weight w belongs to A p(Rn) if

B := sup 1

| B| B w( x) dx1/ p

1

| B| B w1− p′( x) dx

1/ p′

< ∞, (1.6.6)

where the supremum is taken over all balls B in Rn and | B| is a volume of B.

Theorem 1.6.3 ([224], p. 205, p. 210). If conditions (1.6.2) and (1.6.3) are satisfied

and w ∈ A p(Rn) , then the operator K is bounded in L pw(Rn). Further, if (1.6.2) – (1.6.4)

hold and K is bounded in L pw(Rn) , then w ∈ A p(Rn).

The following statements will be useful in Chapter 6.

Lemma 1.6.1 ([224], Section 4.6). Let condition (1.6.3) be satisfied and let u0 be the

unit vector inR

n

. Then by choosing u = tu0 , with t fixed sufficiently large we can guaranteethat

|k (r (u + v)) − k (ru)| ≤ 1

2|k (ru)|

whenever r ∈ R\ 0 and |v| ≤ 2.

Lemma 1.6.2. Let 1 < p < ∞ and let conditions (1.6.3) and (1.6.4) be satisfied. Then

from the boundedness of the operator K given by (1.6.1) from L pw(Rn) to L

pv (Rn) it follows

that w1− p′is locally integrable.

Proof. Let B := B(0,r ). Suppose that I (r ) := B w1− p′( x)dx = ∞ for some positive

number r . Then there exists g ∈ L p( B), g ≥ 0, such that B w−1/ pg = ∞. Let us assume that

f r ( y) = g( y)w−1/ p( y)χ B( y) and B′ = B(ru,r ), where u = tu0 (t is from Lemma 1.6.1 and u0

is the unit vector taken so that (1.6.4) holds). Obviously, x = ru + ux′ for x ∈ B′ and y = ry′

for y ∈ B, where | x′| < 1 and | y′| < 1. Thus x − y = r (u + v) with |v| < 2 and consequently,

Lemma 1.6.1 yields |K f r ( x)| ≥ 12

( f r ) B|k (ru)| for all x ∈ B′. Hence by (1.6.4) the following

estimates hold:

K f r L pv (Rn) ≥ χ B′ ( x)K f r ( x) L

pv (Rn) ≥

b

2rt

(v B′ )1/ p f B = ∞.

On the other hand, f r L pw(Rn) = g L p( B) <∞. Finally we conclude that I (r ) <∞ for all

r > 0.

1.7. Notes and Comments on Chapter 1

Necessary and sufficient conditions guaranteeing the two-weight inequality for the Hardy

operator defined on the semi–axis were derived in [168], [18], [108], [158] (see also the

survey paper [140], monographs [188], [145], [144], [49], [40] and references therein).

For the estimates of the measure of non-compactness for various operators in Lebesgue

spaces we refer to [39], [98], [99], [228], [41], [187], [62], [161], [163], [150] (see also

[40], [49] and references therein).

In the paper [92] it was proved that β( I ) = I , where I is the identity operator from

W k , p

0 (Ω) to L p∗(Ω), k ∈N, 1 ≤ p <∞, k p < n and p∗ = np

n−kp. From this fact it follows that

all entropy numbers of I are equal to I .

Two-sided estimates of entropy and approximation numbers of the embedding operators

between l p and between Sobolev spaces were established in [15], [64], [77], [104], [151],

etc. In the monograph [64] it was shown that if Ω is a bounded domain in Rn and q ∈ [ p, p∗],

where p∗ = npn−kp , then the entropy numbers ek ( I ) of I : W m, p0 (Ω) → Lq(Ω) satisfy ek ( I ) ≈

k −mn .

Lower and upper estimates of a Schatten-von Neumann ideal norms for the Hardy-type

operators were derived in [182], [63], [152] (see also the survey paper [226] and references

therein). This problem for one–sided potentials was studied in [181] and for kernel opera-

tors involving one–sided potentials was investigated in [162] (see also the monograph [49]

and references therein).

For two-sided estimates of singular, entropy and approximation numbers for the Hardy

and potential-type operators we refer to the papers [13], [14], [16], [41], [42], [45], [55],

[65], [175], [149], [67], [181], [34], [82], [83], [164], [49], etc.For estimates of the entropy numbers for the weighted discrete Hardy operators see [25].

In this paper some probabilistic applications of these estimates are also given. The asymp-

totic behaviour of the entropy numbers of diagonal operators generated by logarithmically

decreasing sequences were presented in [20], [146].

The space L p(·) is the special case of the Musielak-Orlicz space. The basis of the vari-

able exponent Lebesgue spaces were developed by W. Orlicz and J. Musielak, H. Hudzik

(see [189], [174], [94]).

The boundedness of the Hardy-Littlewood maximal operator in unweighted L p( x) spaces

was established in the papers [29], [180], [27], [22], [89], [137]. The same problem forclassical integral operators has been investigated in the papers [26], [30], [32], [196], [204],

[205], [120]–[123], [60], [61], [167], [54] (see also [111], [208] and references therein). In

26 Alexander Meskhi

[121] it was shown that if p satisfies the weak Lipschitz condition on Γ , where Γ is a finite

rectifiable curve, then S Γ is bounded in L p( x)(Γ ) if and only if Γ is a Carleson curve, i.e.,

supt ∈Γ , r >0

ν(Γ ∩ B(t ,Γ ))

r < ∞.

The one-weight problem for the Hilbert transform in classical Lebesgue spaces was

solved in [95]. In [24] it was proved that the Calderon- Zygmund singular operator is

bounded in L pw(Rn), 1 < p < ∞, if w ∈ A p(Rn). The necessity of the condition w ∈ A p(Rn)

for the boundedness of R j was established in [73], p. 417 (see also [224] for the related

topics).

The essential norm of the Cauchy integral S T K ( L p(T )), where where T is the unit

circle, has been calculated in [79],[80] for p = 2n and p = 2n

2n−1, where Γ = T is the unit

circle (see also [81]). In these papers a lower estimate for S T K ( L p(T )) has been also

derived for all p ∈ (1,∞). The value of the norm of S T acting in L

p

(T ) (1 < p < ∞) wasfound in [195].

Estimates of the norm for the Ahlfors-Beurling operators and Riesz transforms were

studied in [36]–[38], [96], [10], [177], [192], [193], [194], [35].

Chapter 2

Maximal Operators

This chapter deals with lower estimates of the the measure of non-compactness for the

Hardy–Littlewood and fractional maximal operators in weighted Lebesgue spaces. In par-ticular, we conclude that there is no weight pair for which these operators are compact

from one weighted Lebesgue space into another one. Examples of weight pairs for which

appropriate estimates hold are also given.

2.1. Maximal Functions on Euclidean Spaces

Given any measurable function f on a domain Ω ⊆ Rn we define the maximal operators

M Ω as follows:

M Ω f ( x) = sup B∩Ω∋ x

1| B|

B∩Ω

| f ( y)|dy,

where the supremum is taken over all balls B in Rn with x ∈ B ∩Ω.

If Ω = Rn, then we use the notation

M Ω := M .

The next result is due to B. Muckenhoupt ([169]).

Theorem 2.1.1. Let 1 < p < ∞. Then the operator M is bounded in L pw(Rn) if and only

if w ∈ A p(Rn) (see Definition 1.6.2).

The following statement can be found, e.g., in [73]:

Theorem 2.1.2. Let 1 < p <∞. Then the operator M Ω is bounded in L pw(Ω) if and only

if w ∈ A p(Ω) , i.e.

sup B

1

| B ∩Ω|

B∩Ω

w( x)dx

1/ p 1

| B ∩Ω|

B∩Ω

w1− p′( x)dx

1/ p′

< ∞. (2.1.1)

Now we formulate and prove our main results concerning the maximal operators defined

on Ω.

28 Alexander Meskhi

Theorem 2.1.3 Let 1 < p <∞. Suppose that Ω be a bounded domain in Rn. Then there

is no weight pair (v,w) on Ω for which the operator M is compact from L pw(Ω) to L

pv (Ω).

Moreover, if M Ω is bounded from L pw(Ω) to L

pv (Ω) , then the following estimate holds

M Ωκ ≥ supa∈Ω limr →0

1

| B(a,r )| B(a,r ) v( x)dx1/ p

B(a,r ) w

1− p′

( x)dx1/ p′

.

Proof. It is known (see e.g. [169], [212], [214]) that if M Ω is bounded from L pw(Ω) to

L pv (Ω), then v,w1− p′

∈ Lloc(Ω). Further, let λ > M Ωκ ( L pw(Ω), L

pv (Ω)) and a ∈Ω. By Lemma

1.2.8 we have that there exists a constant β such that if 0 < τ < β, then B(a,τ)∩Ω

v( x) M Ω f

p( x)dx ≤ λ p

Ω

| f ( x)| pw( x)dx. (2.1.2)

Consequently, putting f ( x) = χ B(a,τ)∩Ω( x)w1− p′( x) in (2.1.2) we find that

| B(a,τ)|− p

B(a,τ)

v( x)dx

B(a,τ)

w1− p′( x)dx

p−1

≤ λ p

for all a ∈Ω and sufficiently small τ. The Lebesgue differentiation theorem completes the

proof.

Corollary 2.1.1. Let p = 2 , Ω= (−1,1) , v( x) = w( x) = | x|α , where −1 < α< 1. Then

M Ωκ ≥ 1

(1 −α2)1/2 .

For Ω = Rn we have

Theorem 2.1.4. Let 1 < p < ∞ and let Ω = Rn. Then there is no weight pair (v,w) on

Rn such that M is compact from L

pw(Rn) to L

pv (Rn). Moreover, if M is bounded from L

pw(Rn)

to L pv (Rn) , then the estimate

M κ ≥ max supa∈Rn

limτ→0

I 1(a,τ); supa∈Rn

limτ→∞

I 2(a,τ)

holds, where

I 1(a,τ) = 1

| B(a,τ)|

B(a,τ)

v( x)dx

1/ p B(a,τ)

w1− p′( x)dx

1/ p′

;

I 2(a,τ) = 1

| B(a,2τ)|

B(a,2τ)\ B(a,τ)

v( x)dx

1/ p B(a,τ)

w1− p′( x)dx

1/ p′

.

Proof. Let λ > M κ

. By Lemma 1.2.7 there are constants β1

and β2

such that B(a,τ)

v( x)( M f ( x)) pdx ≤ λ p Rn

| f ( x)| pw( x)dx (2.1.3)

Maximal Operators 29

and Rn\ B(a,t )


| f ( x)| pw( x)dx (2.1.4)

for all τ ≤ β1 and t > β2. Now observe that by (2.1.3) and the arguments used in the proof

of Theorem 1.2.3 we find that

supa∈Rn

limτ→0

I 1(a,τ) ≤ λ,

while (2.1.4) implies that B(a,2t )\ B(a,t )


| f ( x)| pw( x)dx. (2.1.5)

Hence, due to the definition of M and putting f ( x) = χ B(a,t )( x)w1− p′( x) in (2.1.5), we find

that

B(a,2t )\( B(a,t ) v( x)dx B(a,t ) w1− p′

( x)dx p−1

| B(a,2t )|− p

≤ λ p

.

Passing t → ∞ and taking the supremum over all a ∈Rn we derive the desired result.

We say that the measure µ on Rn satisfies a doubling condition ( µ ∈ DC (Rn)) if there

exists a positive constant b such that

µB(a,2r ) ≤ bµ(a,r )

for all a ∈ Rn and r > 0.

Remark 2.1.1. It is known (see e.g. [227], p.21, [236]) that if µ ∈ DC (Rn), then µ ∈ RD(Rn) (reverse doubling condition), i.e. there exist constants η1,η2 > 1 such that

µB(a,η1r ) ≥ η2 µB(a,r )

for all a ∈ Rn and r > 0.

Remark 2.1.2. Analyzing the proof of Theorem 2.1.4 we notice that the constant 2 in

| B(a,2τ)| of the expression I 2(a,τ) might be replaced by some constant σ > 1.

Theorems 2.1.3, 2.1.4 and Remarks 2.1.1 and 2.1.2 yield the next statement.Corollary 2.1.2. Let 1 < p < ∞ and let B be the constant defined by (1.6.6). Suppose

that Ω = Rn and (1.6.6) holds. Then

M K ( L pw(Rn)) ≥ max sup

a∈Rn

limτ→∞

I (a,τ);C B,n supa∈Rn

limτ→0

I (a,τ),

where

I (a,τ) := | B(a,τ)|−1

B(a,τ)

w( x)dx

1/ p B(a,τ)

w1− p′( x)dx

1/ p′

and C B,n is a constant depending only on ¯ B and n.

30 Alexander Meskhi

Proof. By Theorem 2.1.4 it suffices to show that

w( B(a,η1τ) \ B(a,τ)) ≥ cw( B(a,τ)),

where w( E ) :=

E w and the positive constant c depends only on ¯ B and n. But it is easy to

verify that if w ∈ A p(Rn

), then

w( B(a,2τ)) ≤ cn ¯ Bw( B(a,τ))

for all τ > 0 and a ∈Rn. By Remark 2.1.1 we have

w( B(a,η1τ)) \ B(a,τ)) ≥ (η2 − 1)w( B(a,τ)),

where η2 depends only on n and ¯ B.

Corollary 2.1.3. Let 1 < p < ∞ and let Ω be a bounded domain. Suppose that w ∈ A p(Ω) (i.e. (2.1.1) holds ). Then

M Ωκ ( L pw(Ω))

≥ supa∈Rn

limτ→0

| B(a,τ)|−1

B(a,τ)

w( x)dx

1/ p B(a,τ)

w1− p′( x)dx

1/ p′

.

Let us now estimate the measure of non-compactness for the fractional maximal func-

tion

M α,Ω f ( x) = sup

B∩Ω∋ x

1

| B|1−α/n B∩Ω| f ( y)|dy,

where Ω ⊆Rn is a domain.

The next statement is well-known (see [171]):

Theorem 2.1.5. Let 1 < p < ∞ and let 0 < α < n/ p. Suppose that p∗ = npn−α p and

Ω = Rn. Then M α is bounded from L

p

ρ p/ p∗ (Rn) to L p∗

ρ (Rn) if and only if

˜ B := sup B

1

| B|

Bρ( x)dx

1/ p∗ 1

| B|

Bρ− p′/ p∗

( x)dx

1/ p′

< ∞, (2.1.6)

where the supremum is taken over all balls B ⊂ Rn.

The next statement can be derived in the same way as in the case of the Hardy-

Littlewood maximal operator; therefore we omit the proofs.

Theorem 2.1.6. Let 1 < p <∞ , 0 < α< n/ p, p∗ = npn−α p . Suppose that Ω is a bounded

domain in Rn. Then there is no weight pair (v,w) on Ω such that M α,Ω is compact from

L pw(Ω) to L

p∗

v (Ω). Further, suppose that M α,Ω is bounded from L pw(Ω) to L

p∗

v (Ω). Then

M α,Ωκ ( L pw(Ω)→ L

p∗v (Ω))

≥ supa∈Ω

limτ→0

| B(a,τ)|α/n−1

B(a,τ)

v( x)dx

1/ p∗ B(a,τ)

w1− p′( x)dx

1/ p′

.

Corollary 2.1.4. Let 1 < p <∞ , 0 < α< n/ p, p∗ = npn−α p . Suppose that Ω is a bounded

domain in Rn and M α,Ω is bounded from L p

ρ p/ p∗ (Ω) to L p∗

ρ (Ω). Then

M α,Ωκ ( L p

ρ p/ p∗ (Ω)→ L p∗ρ (Ω))

≥ supa∈Ω

limτ→0

| B(a,τ)|α/n−1

B(a,τ)ρ( x)dx

1/ p∗ B(a,τ)ρ− p′/ p∗

( x)dx

1/ p′

.

In the case of Ω = Rn we have the following statement.

Theorem 2.1.7. Let 1 < p <∞ , 0 < α< n/ p, p∗ = npn−α p . Suppose that Ω=R

n and that

(2.1.6) holds. Then there is no weight pair (v,w) such that the operator M α,Ω is compact

L pw(Rn) to L

p∗

v (Rn). Moreover, the following estimate holds:

M α,Ωκ ( L pw(Ω), L p∗

v (Ω)) ≥ max supa∈Rn

limτ→0

I (α)1 (a,τ); supa∈Rn

limt →∞

I (α)2 (a,t ),

where

I (α)1 (a,τ) := | B(a,τ)|α/n−1

B(a,τ)

v( x)dx

1/ p∗ B(a,τ)

w1− p′( x)dx

1/ p′

,

and

I (α)2 (a,τ) := | B(a,2τ)|α/n−1 B(a,2τ)\ B(a,τ)

v( x)dx1/ p∗

B(a,τ)w1− p

′

( x)dx1/ p′

.

Corollary 2.1.5. Let 1 < p < ∞ , 0 < α < n/ p and p∗ = npn−α p . Suppose that ˜ B < ∞ ,

where ˜ B is the constant defined by (2.1.6). Then the inequality

M ακ ( L p

ρ p/ p∗ (Rn), L pρ(Rn)) ≥ max sup

a∈Rn

limτ→0

J (α),C B,n,α, p supa∈Rn

limτ→∞

J (α)(a,τ)

holds, where

J (α)(a,τ) = | B(a,τ)|α/n−1

B(a,τ)ρ( x)dx

1/ p∗ B(a,τ)ρ− p′/ p∗

( x)dx1/ p′

and the positive constant C ˜ B,n,α, p depends on B, n, α and p.

Proof. First observe that the condition (2.1.6) implies ρ ∈ A1+ p∗/ p′ (Rn). Hence the

measure ρ( E ) =

E ρ( x)dx satisfy the doubling condition. Applying Remark 2.1.1, Theo-

rems 2.1.5, 2.1.7 and the arguments from the proof of Corollary 2.1.2 we have the desired

result.

32 Alexander Meskhi

2.2. One–sided Maximal Functions

In this section we deal with the one-sided maximal functions:

N +α f ( x) = sup

h>0

1

h1−α

x+h

x

| f ( y)|dy, x ∈R,

N −α f ( x) = suph>0

1

h1−α

x

x−h| f ( y)|dy, x ∈R,

where 0 ≤ α < 1. If α = 0, then we denote N + := N +0 and N − := N −0 .

Definition 2.2.1. Let 1 0

such that

1

h

x

x−h

w(t )dt

1/ p

1

h

x+h

x

w1− p′(t )dt

1/ p′

≤ c; x ∈ R, h > 0.

Further, w ∈ A− p (R) if

1

h

x+h x

w(t )dt

1/ p1

h

x x−h

w1− p′(t )dt

1/ p′

≤ C ; x ∈ R, h > 0,

for some positive constant C .

Theorem 2.2.1 ([216], [3]). Let 1 < p < ∞. Then

(i) N + is bounded in L p

w(R) if and only if w

∈ A+

p (R);

(ii) N − is bounded in L pw(R) if and only if w ∈ A−

p (R).

Definition 2.2.2. Let p and q be constants such that 1 < p <∞, 1 < q <∞. We say that

a weight ρ ∈ A+ pq(R) if

sup0<h≤ x

1

h

x x−h

ρq(t )dt

1q

1

h

x+h x

ρ− p′(t )dt

1 p′

< ∞.

Further, ρ ∈ A− pq(R+) if

sup0<h≤ x

1

h

x+h x

ρq(t )dt

1q

1

h

x x−h

ρ− p′(t )dt

1 p′

< ∞.

Theorem 2.2.2 ([3]). Suppose that 0 < α < 1 , 1 < p < 1α and q = p

1−α p . Then N +α is

bounded from L pρ p (R) t o L

qρq (R) if and only if ρ ∈ A+

pq(R). Further, N −α is bounded from

L pρ p (R) to L

qρq (R) if and only if ρ ∈ A−

pq(R).In the next statement we assume that the symbol N α denotes one of the operators N +α ,

N −α .

Theorem 2.2.3. Let 1 < p ≤ q < ∞ ,and α = 1/ p − 1/q. Then there is no weight pair

(v,w) defined on R for which the operator N α is compact from L pw(R) to L

qv (R). Moreover,

if N α is bounded from L pw(R) to L

qv (R) for some weight functions v and w, then

N +

ακ

≥ 2α−1 supa∈R

limh→01

h a

a−hv( x)dx

1/q

1

h a+h

aw1− p′

( x)dx1/ p′

;

N −α κ ≥ 2α−1 supa∈R

limh→0

1

h

a+h

av( x)dx

1/q1

h

a

a−hw1− p′

( x)dx

1/ p′

.

Proof. We prove the theorem for N α ≡ N +α . Let λ > N +α κ . First observe that the

boundedness of N +α from L pw(R) to L

qv (R) implies

J (t ) := t

−t w1− p′

(τ)d τ < ∞

for every t > 0. Indeed, if J (t ) = ∞ for some t > 0, then there is a non-negative function g

on (−t ,t ) belonging to L p([−t , t ]) such that t

−t g(τ)w−1/ p(τ)d τ = ∞.

Assuming now that f t ( y) = g( y)w−1/ p( y)χ(−t ,t )( y) we find that

N +α f t Lqv (R) ≥ χ(−2t ,−t ) N +α f t L

qv (R)

≥ ct α−1 −t

−2t v( x)dx1/q t

−t g( y)w−1/ p( y)dy = ∞

On the other hand,

f t L pw(R) =

t

−t g p( x)dx < ∞

which contradicts the boundedness of N +α from L pw(R) to L

qv (R). Consequently, we conclude

that J (t ) < ∞ for every t > 0.

Repeating the arguments of the proof of Theorem 2.1.4 we have that

a+τ

a−τv( x)( N +α f ( x))qdx ≤ λq

R| f ( x)| pw( x)dxq/ p

.

for all a ∈ R and small τ, where λ > N +α κ .Hence

1

(2τ)1−α

a

a−τv( x)dx

a+τ

a f (t )dt

q

dx ≤ λq

R

f p( x)w( x)dx

q/ p

, (2.2.1)

where f ≥ 0.

Assuming f (t ) = χ(a,a+τ)(t )w1− p′(t ) in (2.2.1), passing now to the limit when τ → 0

and taking the supremum with respect to all a we have the desired estimate for N +α . TheLebesgue differentiation theorem completes the proof.

The proof for N −α is similar to that for N +α .

34 Alexander Meskhi

2.3. Maximal Operator on Homogeneous Groups

Let G be a homogeneous group and let

M α f ( x) = sup

B∋ x

1

| B|1− α

Q B

| f ( y)|dy, x ∈ G, 0 ≤ α < Q,

where the supremum is taken over all balls B ⊂ G containing x. If α = 0 then M α becomes

the Hardy-Littlewood maximal function which will be denoted by M .

The following statements hold (see, e.g., [70], [76]).

Theorem 2.3.1. Let 1 < p < ∞. Then the Hardy-Littlewood maximal function M is

bounded in L pρ (G) if and only if ρ ∈ A p(G) i.e.

sup B 1

| B| B

ρ( x)dx1 p 1

| B| B

ρ1− p′

( x)dx 1

p′< ∞, (2.3.1)

where the supremum is taken over all balls B ⊂ G.

Theorem 2.3.2. Let 1 < p < ∞ , 0 < α < Q p

. Then the fractional maximal function M α

is bounded from L pρ p (G) to L

qρq (G) , where q = Qp

Q−α p , if and only if ρ ∈ A pq(G) i.e.

sup B

1

| B|

B

ρq( x)dx 1

q 1

| B|

B

ρ− p′

( x)dx 1

p′< ∞. (2.3.2)

Now we formulate and prove the main results of this subsection.

Theorem 2.3.3. Let 1 < p <∞. Suppose that M is bounded from L pw(G) to L

pv (G). Then

there is no weight pair (v,w) such that M is compact from L pw(G) to L

pv (G). Moreover, the

inequality

M κ ( L pw(G), L p

v (G)) ≥ supa∈G

limτ→0

1

| B(a,τ)|

B(a,τ)

v( x)dx 1

p

B(a,τ)

w1− p′

( x)dx 1

p′

holds.

Proof. Suppose that λ > M κ ( L pw(G)→ L

pv (G)) and a ∈ G. By Lemma 1.2.7 we have that

B(a,τ)

v( x)( M f ( x)) pdx ≤ λ p

B(a,τ)

| f ( x)| pw( x)dx (2.3.3)

for all τ (τ≤ β) and all f supported in B(a,τ). First observe that

B(a,τ)

w

1− p′

( x)dx < ∞

for all τ > 0 (see also, for example, [168], [227], [76], Ch. 4).

Further, substituting f ( y) = χ B(a,r )( y) w1− p′

( y) in inequality (2.3.3)we find that

1

| B(a,τ)| p

B(a,τ)

v( x)dx

B(a,τ)

w1− p′

( x)dx p−1

≤ λ p.

This inequality and Lebesgue differentiation theorem for homogeneous groups (see

[70], p. 67) yield the desired result.

Theorem 2.3.4. Let 1 < p < ∞ , 0 < α < Q/ p and let q = QpQ−α p . Suppose that M α is

bounded from L pw(G) to L

qv (G). Then there is no weight pair (v,w) such that M α is compact

from L pw(G) to L

qv (G). Moreover, the inequality

M ακ ≥ supa∈G

limτ→0

1

| B(a,τ)|αQ

−1

B(a,τ)

v( x)dx 1

q

B(a,τ)

w1− p′

( x)dx 1

p′

holds.

The proof of this statement is similar to that of Theorem 2.3.3; therefore it is omitted.

Example 2.3.1. Let 1 < p < ∞, v( x) = w( x) = r ( x)γ , where −Q < γ < ( p − 1)Q. Then

M κ ( L pw(G)) ≥ Q

(γ + Q)

1 p (γ (1 − p

′) + Q)

1

p′ −1

.

Indeed, first recall that the fact that | B(e,1)| = 1 and Proposition 1.1.1 impliesσ(S ) = Q,

where S is the unit sphere in G and σ(S ) is its measure. Taking into account Theorems 2.3.1and 2.3.3 we have

M κ ( L pw(G)) ≥ lim

τ→0

1

| B(e,τ)|

B(e,τ)

w( x)dx

1/ p B(e,τ)

w1− p′( x)dx

1/ p′

= σ(S ) limτ→0τ−Q

τ0

t γ +Q−1dt

1/ p τ0

t γ (1− p′)+Q−1dt

1/ p′

= Q(γ + Q)1 p (γ (1 − p

′) + Q)

1

p′

−1

.


This chapter is based on the results of the papers [58], [43], [5].

A result analogous to that of [58] has been obtained in [184], [185] for the Hardy-

Littlewood maximal operators with more general differentiation bases on symmetric spaces.

The one-weight problem for the Hardy-Littlewood maximal functions was solved by

B. Muckenhoupt [169] (for maximal functions defined on quasimetric measure spaces with

doubling condition see, e.g., [227]) and for fractional maximal functions and Riesz poten-

tials by B. Muckenhoupt and R. L. Wheeden [171]. Theorem 2.3.2 for Euclidean spaces

was derived in [171] and for homogeneous groups and quasimetric measure spaces with

doubling condition, for instance, in [70], Ch. 6; [76], Ch. 4.

36 Alexander Meskhi

Two-weight criteria involving the operator itself for the Hardy-Littlewood maximal op-

erator have been established in [212] (see [172], [191] for another type of sufficient condi-

tions).

The two-weight problem for the fractional maximal functions has been solved in [212],

[214], [217], [236] (see also [78], [76], Ch. 4, for quasimetric measure spaces).

Sharp estimates for the Hardy-Littlewood maximal functions were obtained in [84],[85]. In [19] the author found the sharp dependence of M L

pw(Rn) on B (see (1.6.6) for the

definition of B ) in Theorem 2.1.2 for Ω=Rn. In particular, if 1 < p <∞, then M L

pw(Rn) ≤

C p,n( B) p′and the exponent p′ is the best possible. This result was used in [35] to establish

sharp one-weighted estimates for the Hilbert, Beurling and martingale transforms.

Chapter 3

Kernel Operators on Cones

Let E be a cone in a homogeneous group G (see Section 1.1 for the definition). We denote

E t := y ∈ E : r ( y) < t .

In the sequel we also use the notation:

S x := E r ( x)/2c0, F x := E r ( x)\S x,

where the constant c0 comes from the triangle inequality for r (see Section 1.1).

In this chapter boundedness/compactness criteria from L p( E ) to Lqv ( E ) are established

for the operator

K f ( x) = E r ( x)

k ( x, y) f ( y)dy, x ∈ E , ( A)

with positive kernel k , where 1 < p,q < ∞ or 0 < q ≤ 1 < p < ∞, E r ( x) and E are certain

cones in homogeneous groups and k satisfies conditions which in the one-dimensional case

are similar to those of [163]. The measure of non-compactness for K is also estimated from

the both sides.

We present also two-sided estimates of Schatten-von Neumann norms for the operator

with positive kernel

Ku f ( x) = u( x) E r ( x)

k ( x, y) f ( y)dy, x ∈ E , ( B)

where u is a measurable function on E .

We need some definitions regarding the kernel k .

Definition A. Let k be a positive function on ( x, y) ∈ E × E : r ( y) < r ( x) and let

1 < λ < ∞. We say that k ∈ V λ , if there exist positive constants c1 , c2 and c3 such that

(i) k ( x, y) ≤ c1k ( x,δ1/(2c0) x) (C )

for all x, y ∈ E with r ( y) < r ( x)/(2c0);

k ( x, y) ≥ c2k ( x,δ1/(2c0) x) (C ′)

38 Alexander Meskhi

for all x, y ∈ E with r ( x)/(2c0) < r ( y) < r ( x);

(ii) F x

k λ′( x, y)dy ≤ c3r Q( x)k λ

′( x,δ1/(2c0) x), λ′ = λ/(λ− 1), ( D)

for all x ∈ E.Example A. Let G = Rn, r ( xy−1) = | x − y|, δt x = tx, x, y ∈ Rn. If k ( x, y) = | x − y|α−n,

then k ∈ V λ, where n/λ < α ≤ n. Indeed, it is easy to see that if y ∈ S x, then | x| ≤ | x − y| +| y| ≤ | x − y| + | x|/2. Hence | x|/2 = k ( x, x/2) ≤ k ( x, y). Consequently, (C ) holds. Further, it

is easy to see that (C ′) is also satisfied. Moreover, we have

F x

| x − y|(α−n)λ′ dy = ∞

0

y ∈ F x : | x − y|(α−n)λ′ > εd ε

≤ | x|(α−n)λ′

0

(· · · ) +

∞

| x|(α−n)λ′

(· · · ) := I1 + I2.

For I1 we have

I1 ≤

| x|(α−n)λ′ 0

B(0, | x|)d ε = c| x|(α−n)λ′+n,

while for I2 we observe that

I2 ≤

∞

| x|(α−n)λ′ y : | y| < | x|, | x − y| ≤ ε1/(α−n)λ′

d ε

≤ c

∞ | x|(α−n)λ′

εn/(α−n)λ′ d ε = cα,n

| x|(α−n)λ′

1+n/(α−n)λ′

= cα,n| x|n+(α−n)λ′.

Example B. It is easy to see that if the following two conditions are satisfied for k :

(i) k (δt ¯ x,δτ ¯ y) ≤ c1k (δt ¯ x,δs ¯ z)

for all t , τ, s, ¯ x, ¯ y, ¯ z with 0 < τ < s < t ; ¯ x, ¯ y, ¯ z ∈ A;

(ii)

t

t /(2c0)k λ

′(δt ¯ x,δτ ¯ y)τQ−1d τ≤ c2t Q · k λ

′(δt ¯ x,δt /(2c0) ¯ x), t > 0, ¯ x ∈ A,

then k ∈ V λ.

Example C. Let k ( x, y) = k (r ( x),r ( y)) be a radial kernel. It is easy to check that if there

exist positive constants c1 and c2 such that

(i) k (s, l) ≤ c1k (s,t ), 0 < l < t < s,

(ii) t

t /(2c0)k λ

′(t ,s)sQ−1ds ≤ c2t Qk λ

′(t ,t /(2c0)), 1 < λ < ∞,

then k ∈ V λ.

Kernel Operators on Cones 39

3.1. Boundedness

In this section we establish boundedness/compactness criteria for K acting from L p( E ) to

Lqv ( E ), 1 < p ≤ q < ∞.

Theorem 3.1.1. Let 1 < p ≤ q < ∞ and let v be a weight on E. Suppose that k ∈ V p.

Then K is bounded from L p( E ) to Lqv ( E ) if and only if

B := sup j∈Z

B( j) := sup j∈Z

E

2 j+1 \ E 2 j

v( x)k q( x,δ1/(2c0) x)dx1/q

2 jQ/ p′

< ∞. (3.1.1)

Proof. Sufficiency. Let f ≥ 0 on E . We have

K f q

Lqv ( E )

≤ c

E

v( x)

S x

k ( x, y) f ( y)dyq

dx +

E

v( x)

F x

k ( x, y) f ( y)dyq

dx

:= c(I1 + I2).

By condition (C) and Theorem 1.3.2 we find that

I1 ≤ c

E

v( x)k q( x,δ1/(2c0) x)

E r ( x)

f ( y)dyq

dx ≤ c

E

f p ( x)dxq/ p

,

while Holder’s inequality and condition (D) yield

I2 ≤

E v( x)

F x

k p′( x, y)dy

q/ p′ F x

f p ( y)dyq/ p

dx

≤ c E

v( x)k q( x,δ1/(2c0) x)r Qq/ p′ ( x) F x

f p ( y)dyq/ p

dx

≤ c∑ j∈Z

E

2 j+1 \ E 2 j

v( x)k q( x,δ1/(2c0) x)r Qq/ p′( x)

F x

f p ( y)dyq/ p

dx

≤ c∑ j∈Z

E

2 j+1 \ E 2 j

v( x)k q( x,δ1/(2c0) x)r Qq/ p′( x)dx

× E

2 j+1 \ E 2 j−1/c0

f

p

( y)dyq/ p

≤ cB

q

f

q

L p( E ).

Sufficiency has been proved.

Necessity. To prove necessity we take the functions f j( x) = χ E 2 j+1

( x). Then simple

calculations show that f j L p( E ) = c2 jQ/ p. Further, condition (C) yields

K f j( x)q

Lqv ( E )

≥

E 2 j+1 \ E

2 j

v( x)

F x

f j( y)k ( x, y)dyq

dx

≥ c E

2 j+1 \ E 2 j

v( x)k q

( x,δ1/(2c0) x)dx2 jQq

.

Finally, due to the boundedness of K we have the desired result.

40 Alexander Meskhi

Now we consider the case q < p.

Theorem 3.1.2. Let 0 < q 1. Suppose that k ∈ V p. Then the

following conditions are equivalent:

(i) K is bounded from L p( E ) to Lqv ( E );

(ii)

D :=

E

E \ E r ( x)

k q( y,δ1/(2c0) y)v( y)dy

p/( p−q)

r ( x)Qp(q−1)/( p−q)dx

( p−q)/ pq

< ∞.

Proof. First we show that (ii) ⇒ (i). Suppose that f ≥ 0. Keeping the notation of

sufficiency of the proof of Theorem 3.1.1, using Proposition 1.1.2 and condition (C) we

have

I1 ≤ c E v( x)k q( x,δ1/(2c0) x) S

x

f ( y)dyq

dx

= c

∞0

V (t )

t /(2c0)

0F (τ)d τ

q

dt ,

where

V (t ) := t Q−1

Av(δt ¯ x)k q(δt ¯ x,δt /(2c0) ¯ x)d σ( ¯ x),

F (t ) := t Q−1

A

F (δt ¯ x)d σ( ¯ x).

Notice that

D = ∞

0

∞t

V (τ)d τ p/( p−q)

t Qp(q−1)/( p−q)+Q−1dt ( p−q)/( pq)

= c

∞0

∞t

V (τ)d τ

p/( p−q)

×

t

0τ(Q−1)(1− p)(1− p′)d τ

p(q−1)/( p−q)

t (Q−1)(1− p)(1− p′)dt

( p−q)/( pq)

.

Consequently, due to Theorem 1.3.3, Holder’s inequality and Proposition 1.1.2 we find

that

I1 ≤ c

∞0

t (Q−1)(1− p)F p(t )dt

q/ p

= c

∞0

t (Q−1)(1− p)

A

f (δt ¯ x)d σ( ¯ x)

p

t (Q−1) pdt

q/ p

≤ c

∞0

t Q−1

A

f p(δt ¯ x)d σ( ¯ x)

dt

q/ p

= c

E

f p( x)dx

q/ p

.

Further, applying Holder’s inequality twice and condition (D) we find that

I2 ≤

E v( x)

F x

f p ( y)dy

q/ p F x

k p′( x, y)dy

q/ p′

dx

≤ c

E

v( x)

F x

f p( y)dy

q/ p

r Qq/ p′( x)k q( x,δ1/(2c0) x)dx

= c∑ j∈Z

E

2 j+1 \ E 2 j

v( x)

F x

f p( y)dy

q/ p

r ( x)Qq/ p′k q( x,δ1/(2c0) x)dx

≤ c∑ j∈Z

E

2 j+1 \ E 2 j−1/c0

f p ( y)dy

q/ p E

2 j+1 \ E 2 j

v( x)r Qq/ p′( x)k q( x,δ1/(2c0) x)dx

≤ c

∑

j∈Z

E

2 j+1 \ E 2 j−1/c0

f p( y)dy

q/ p

×

∑

j∈Z

E

2 j+1 \ E 2 j


p/( p−q)( p−q)/ p

≤ c f q

L p( E )

∑

j∈Z

E

2 j+1 \ E 2 j


p/( p−q)( p−q)/ p

=: c f q

L p( E )( ¯ D)q.

Besides this,

( ¯ D) pq/( p−q)≤c∑ j∈Z

2Qq( p−1) j/( p−q)

E

2 j+1 \ E 2 j

v( x)k q( x,δ1/(2c0) x)dx

p/( p−q)

≤ c∑ j∈Z

E

2 j \ E 2 j−1

r ( y)Qp(q−1)/( p−q)

×

E \ E r ( y)


p/( p−q)

dy

≤ cD pq/( p−q) < ∞.

Now we show that (i)⇒(iii). Let vn( x) = v( x)χ E n\ E 1/n( x), where n is an integer with

n ≥ 2. Let

f n( x) =

E \ E r ( x)

vn( z)k q( z,δ1/(2c0) z)dz

1/( p−q)

r ( x)Q(q−1)/( p−q).

It is easy to see that

f L p( E ) =

E

E \ E r ( x)

k q( y,δ1/(2c0) y)vn( y)dy

p/( p−q)

×r ( x)Qp(q−1)/( p−q)dx1/ p

< ∞.



42 Alexander Meskhi

On the other hand, by condition (C ′) and integration by parts we find that

K f Lqv ( E ) ≥

E

vn( x)

F x

f n( y)k ( x, y)dy

q

dx

1/q

≥ c E

vn( x)k q( x,δ1/(2c0) x) F x

f n( y)dyq

dx1/q

= c

E

vn( x)k q( x,δ1/(2c0) x)

F x

E \ E r ( y)


1/( p−q)

×r ( y)Q(q−1)/( p−q)dy

q

dx

1/q

≥ c E vn( x)k q( x,δ1/(2c0) x) E \ E

r ( x)

vn( z)k q( z,δ1/(2c0) z)dzq/( p−q)

×

F x

r ( y)Q(q−1)/( p−q)dy

q

dx

1/q

≥ c

E

vn( x)k q( x,δ1/(2c0) x)

E \ E r ( x)


q/( p−q)

×r ( x)Qq( p−1)/( p−q)dx

1/q

c ∞0

t Q−1 A

vn(δt ¯ x)k q(δt ¯ x,δt /(2c0) ¯ x)d σ( ¯ x) ∞t τQ−1

×

A

vn(δτ ¯ z)k q(δτ ¯ z,δτ/(2c0) ¯ z)d σ(¯ z)

d τ

q/( p−q)

t Qq( p−1)/( p−q)dt

1/q

= c

∞0

d

∞t τQ−1

A

vn(δτ ¯ x)k q(δτ ¯ x,δτ/(2c0) ¯ x)d σ( ¯ x)

d τ

p/( p−q)

×t Qq( p−1)/( p−q)dt

1/q

= c

∞

0 ∞

t τQ−1

Avn(δτ ¯ x)k q(δτ ¯ x,δτ/(2c0))d σ( ¯ x)

d τ

p/( p−q)

×t Qq( p−1)/( p−q)dt 1/q

= c

∞0

t Q−1

∞t τQ−1

A

vn(δτ ¯ x)k q(δτ ¯ x,δτ/(2c0) ¯ x)d σ( ¯ x)

d τ

p/( p−q)

×t Qq( p−1)/( p−q)−Qdt

1/q

= c E E \ E r ( x)

vn( x)k q( x,δ1/(2c0) x)dx p/( p−q)

r ( x)Qp(q−1)/( p−q)dx1/q

.

Now the boundedness of K and Fatou’s lemma completes the proof of the implication

(i)⇒ (ii).

3.2. Compactness

This section deals with the compactness of the operator K.

Theorem 3.2.1. Let 1 < p ≤ q < ∞. Suppose that k ∈ V p. Then K is compact from

L p( E ) to Lqv ( E ) if and only if

(i) (3.1.1) holds;(ii)

lim j→−∞

B( j) = lim j→+∞

B( j) = 0 (3.2.1)

(see (3.1.1) for B( j)).

Proof. Sufficiency. Let 0 < a < b < ∞ and represent K f as follows:

K f = χ E aK ( f χ E a ) +χ E b\ E aK( f χ E b )

+χ E \ E bK( f χ E b/2c0 ) +χ E \ E bK( f χ E \ E b/2c0 )

:= P1 f + P2 f + P3 f + P4 f .

For P2 we have

P2 f ( x) = E

k ∗( x, y) f ( y)dy,

where k ∗( x, y) = χ E b\ E a ( x)χ E r ( x)( y)k ( x, y). Further, observe that

S := E

E k ∗( x, y) p′

dyq/ p′

v( x)dx = E b\ E a

E r ( x)

k p′( x, y)dy

q/ p′

v( x)dx

≤ c

E b\ E a

S x

k p′( x, y)dy

q/ p′

v( x)dx + c

E b\ E a

F x

k p′( x, y)dy

q/ p′

v( x)dx

:= S 1 + S 2.

Taking into account the condition k ∈ V p we have

S i ≤ c E b\ E a

k q

( x,δ1/(2c0) x)r qQ/ p′

( x)v( x)dx

≤ cbq/ p′

E b\ E a

k q( x,δ1/(2c0) x)v( x)dx < ∞, i = 1,2.

Finally we have S < ∞ and consequently, by Theorem 1.3.5 we conclude that P2 is

compact for every a and b. Analogously, we obtain the compactness of P3.

Further, by the arguments used in the proof of Theorem 3.1.1 we have

P1 ≤ cB(a)

:= c supt ≤a E a\ E t

v( x)k q

( x,δ1/(2c0) x)dx1/q

t Q/ p′

;

44 Alexander Meskhi

P4 ≤ cB(b) := c supt ≥b

E \ E t

v( x)k q( x,δ1/(2c0) x)dx1/q

t Q − bQ1/ p′

.

Therefore

K− P2 − P3 ≤ P1 + P4 ≤ c B(a) + B(b).It remains to show that condition (3.2.1) implies

lima→0

B(a) = limb→∞

B(b) = 0.

Let a > 0. Then a ∈ [2m,2m+1) for some integer m. Consequently, B(a) ≤ B(2m+1). Further,

for 0 < t < 2m+1, there exists j ∈ Z , j ≤ m, such that t ∈ [2 j,2 j+1). Hence

t qQ/ p′

E 2m+1 \

E t

v ( x)k q( x,δ1/(2c0) x))dx

≤ 2( j+1)qQ/ p′ m

∑k = j

E

2k +1 \ E 2k


≤ cBq(k )2 jqQ/ p′ ∞

∑k = j

2−kqQ/ p′= cBq(k ).

Taking into account this estimate we shall find that

B(2m) ≤ supk ≤m

( B(k ))q =: S (m).

If a → 0, then m → −∞. Consequently, S (m) → 0, which implies that B(2m) → 0 as a → 0.

Finally we have that B(a) → 0 as a → 0.

Now we take arbitrary b > 0. Then b ∈ [2m,2m+1) for some m ∈ Z . Hence B(b) ≤ B(2m).

Further, for t > 2m, there exists k ≥ m, k ∈ Z such that t ∈ [2k ,2k +1). For such a t we have E \ E t

v( x) k q ( x,δ1/(2c0) x)dx

t Q − 2mQq/ p′

≤ E \ E

2k

v( x)k q( x,δ1/(2c0) x)dx2(k +1)Q − 2mQq/ p′

≤ c2kqQ/ p′ ∞

∑ j=k

E

2 j+1 \ E 2 j

v( x)k ( x,δ1/(2c0) x)q( x)dx

≤ cBq(k )2kqQ/ p′ ∞

∑ j=k

2− jqQ/ p′≤ cBq(k ).

Consequently, B(2

m

) ≤ supk ≥m

B(

k )

. From the last inequality we conclude that the condition

limk →+∞

B(k ) = 0 implies limb→∞

B(b) = 0.

Necessity. Let K be compact. As (3.1.1) is obvious we have to prove (3.2.1). Let j ∈ Z

and put f j( y) = χ E 2 j+1 \ E

2 j−1/c0

( y)2− jQ/ p. Then for φ ∈ L p′( E ), we have

E

f j( y)φ( y)dy

≤

E 2 j+1 \ E 2 j−1/c0

| f j( y)| pdy

1/ p

E 2 j+1 \ E 2 j−1/c0

|φ( y)| p′dy

1/ p′

=

E 2 j+1 \ E

2 j−1/c0

|φ( y)| p′dy 1

p′

→ 0

as j → −∞ or j → +∞. On the other hand,

K f j Lqv ( E ) ≥

E

2 j+1 \ E 2 j

K f j( x)

qv( x)dx

1/q

≥ c E

2 j+1 \ E 2 j

F x

k ( x, y) f j( y)dyq

v( x)dx1/q

≥ c E

2 j+1 \ E 2 j

k q( x,δ1/(2c0) x)

F x

f j( y)dyq

v( x)dx1/q

≥ c

E 2 j+1 \ E

2 j

k q( x,δ1/(2c0) x)v( x)dx1/q

2 jQ/ p′= cB( j).

As a compact operator maps a weakly convergent sequence into a strongly convergent one,

we conclude that (3.2.1) holds.

For q 1. Suppose that k ∈ V p. Then the


(i) K is compact from L p( E ) to Lqv ( E );

(ii)

D :=

E

E \ E r ( x)

k q( y,δ1/(2c0) y)v( y)dy p/( p−q)

r ( x)Qp(q−1)/( p−q)dx( p−q)/ pq

< ∞.

Proof follows immediately from Theorems 3.1.2 and 1.3.6.

3.3. Schatten–von Neumann norm Estimates

In this section we give necessary and sufficient conditions guaranteeing two-sided estimates

of the Schatten-von Neumann norms for the operator given by (B).

We denote by k 0 the function r ( x)Qk 2( x,δ1/(2c0) x).

46 Alexander Meskhi

Theorem 3.3.1. Let 2 ≤ p < ∞ and let k ∈ V 2. Then Ku ∈ σ p( L2( E )) if and only if

u ∈ l p( L2k 0

( E )). Moreover, there exist positive constants b1 and b2 such that

b1ul p( L2k 0

( E )) ≤ Kuσ p( L2( E )) ≤ b2ul p( L2k 0

( E )).

Proof. Sufficiency. First observe that

J ( x) :=

E r ( x)

k 2( x, y)dy ≤ ck 0( x),

where c is a positive constant. Indeed, splitting the integral over E r ( x) into two parts and

taking into account the condition k ∈ V 2 we have

J ( x) =

S x

k 2( x, y)dy +

F x

k 2( x, y)dy ≤ c1k 0( x) + c2k 0( x) ≤ c3k 0( x).

Hence, the Hilbert-Schmidt formula yields

Kuσ2( L2( E )) =

E × E r ( x)

k ( x, y)u( x)

2dxdy

1/2

=

E

u2( x)

E r ( x)

k 2( x, y)dy

dx1/2

≤ c

E

u2( x)k 0( x)dx1/2

= c∑n∈Z

E

2n+1 \ E 2n

u2( x)k 0( x)dx1/2

= cul2( L2k 0

( E )).

On the other hand, by Theorem 3.1.1 and Proposition 1.5.1 we have that there is a positive

constants c such that

Kuσ p( L2( E )) ≤ cul p( L2k 0

( E )),

where 2 ≤ p < ∞.

Necessity. Let Ku ∈ σ p

L2( E )

. We set

f n( x) = χ E 2n+1 \ E 2n ( x)| E 2n+1 \ E 2n |− 12 ;

gn( x) = u( x)χ E 2n+1 \ E

3·2n−1( x)k

1/20 ( x)α

−1/2n ,

where

αn =

E

2n+1 \ E 3·2n−1

u2( x)k 0( x)dx.

Then it is easy to verify that f n and gn are orthonormal systems in L2( E ). By Proposi-

tion 1.5.2 we find that

∞ > Kuσ p( L2( E )) ≥∑

k ∈Z

<Ku f n, gn > p1/ p

= ∑n∈Z E 2n+1 \ E 3·2n−1

k 1/2

0 ( x)(K f n)( x)u2( x)α

−1/2n dx

p

1/ p

≥ c∑

n∈Z

E

2n+1 \ E 3·2n−1

k 1/20 ( x)u2( x)α

−1/2n

E r ( x)\ E 2n

k ( x, y) f n( y)dy

dx p1/ p

≥ c∑

n∈Z

E

2n+1 \ E 3·2n−1

k ( x,δ1/(2c0) x)u2( x)α−1/2n 2−nQ/2(r ( x)Q− 2nQ)dx

p1/ p

≥ c∑

n∈Z

E

2n+1 \ E 3·2n−1

u2( x)k 0( x)α−1/2n dx

p1/ p

= c ∞∑

n=0

α p/2n

1/ p

.

Let us now take

f ′n( x) = χ E 3·2n−1 \ E

3·2n−2( x)| E 3·2n−1 \ E 3·2n−2 |− 1

2 ;

g′n( x) = u( x)χ E

3·2n−1 \ E 2n ( x)k 1/20 ( x)β

−1/2n ,

where

βn =

E

3·2n−1 \ E 2n

u2( x)k 0( x)dx

and argue as above. Then we conclude that

∞ > Kuσ p

L2( E )

≥ c ∞∑

n=0

β p/2n

1/ p

.

Summarizing the estimates obtained above we have∑

n∈Z

E

2n+1 \ E 2n

u2( x)k 0( x)dx

p/21/ p

≤

∑

n∈Z

(αn +βn) p/2

1/ p

≤ c(Kuσ p( L2(0,∞)) + Kuσ p( L2(0,∞)))

≤ 2cKuσ p( L2(0,∞)).


In this section we present two-sided estimates for the measure of non-compactness

KK ( L p( E ), Lqv ( E )) of the operator K.

Theorem 3.4.1. Let 1 < p ≤ q <∞ and let k ∈ V p. Assume that K is bounded from X to

Y, where X = L p

( E ) and Y = L

q

v ( E ). Then there exist positive constants b1 (depending onc1 , c3 , p and q) and b2 (depending on c2 p and q) such that the inequality

b1 J ≤ KK ( X ,Y ) ≤ b2 J

48 Alexander Meskhi

holds, where

J = lim j→+∞

B( j) + lim j→−∞

B( j),

B( j) is defined by (3.1.1) and the constants c1 , c2 and c3 are defined in Definition A.

Proof. From the proof of Theorem 3.2.1 we see that

K− P(n,m)1 − P

(n,m)2 ≤ b2

sup j≥m

B( j) + sup j≤n

B( j),

where P(n,m)1 and P

(n,m)2 are compact operators for every n,m ∈ Z, n < m. Consequently,

KK ( X ,Y ) ≤ b2 J ,

where b2 depends only on p, q, c1 and c3.

To obtain the lower estimate

KK ( X ,Y ) ≥ b1 J ,

we take λ > KK ( X ,Y ). Then by Lemma 1.2.3 there exists P ∈ F L( X ,Y ) such that K−P < λ. On the other hand, using Lemma 1.2.6, for ε = (λ− K− P)/2, there exist

T ∈ F L( X ,Y ) and E α,β := x ∈ E : 0 < α < r ( x) < β < ∞ such that

P − T < ε (3.4.1)

and

supp T f ⊂ E α,β.

From (3.4.1) we obtain

K f − T f Y ≤ λ f X

for every f ∈ X . Thus, E α

|K f ( x)|qv( x)dx +

E \ E β

|K f ( x)|qv( x)dx ≤ λq f q

X (3.4.2)

for every f ∈ X .

Let us choose n ∈ Z such that 2n < α. Assume that j ∈ Z , j ≤ n and f j( y) = χ E 2 j+1

.

Then using condition (C ′) we find that

E 2 j+1 \ E

2 j

|K f j( x)|qv( x)dx ≥

E 2 j+1 \ E

2 j

F x

k ( x, y) f j( y)dy

q

v( x)dx

≥ c

E

2 j+1 \ E 2 j

k q( x,δ1/(2c0) x)v( x)r ( x)qdx.

On the other hand, f jq

X = c2 jQq/ p and consequently, (3.4.2) yields

cB( j) ≤ λ

for every integer j, j ≤ n. Hence sup j≤n B( j) ≤ cλ for all integers n with the condition

2n < α. Therefore limn→−∞

sup j≤n B( j) ≤ cλ.

Now we take m ∈ Z such that 2m > β. Then for f j( y) = χ E 2 j+1)

( y) ( j ≥ m), we obtain

E

2 j+1 \ E 2 j

|K f j( x)|q

v( x)dx ≥ c E

2 j+1 \ E 2 j

k q

( x,δ1/(2c0) x)v( x)r ( x)q

dx.

On the other hand, f jq

X = c2Q jq/ p. Hence sup j≥m B( j) ≤ cλ, where c depends only on p,

q and c1. Consequently, limm→+∞

sup j≥m B( j) ≤ cλ from which it follows the desired estimate.

3.5. Convolution–type Operators with Radial Kernels

Let ϕ be a positive function on [0,∞) and let

K f ( x) =

E r ( x)

ϕ(r ( xy−1)) f ( y)dy, x ∈ E .

We say that ϕ belongs to U λ, 1 < λ < ∞, if

(a) there exists a positive constants c1 and c2 such that

ϕ(r ( xy−1)) ≤ c1ϕ(r ( x)), y ∈ S x,

ϕ(r ( xy−1

)) ≥ c2ϕ(r ( x)), y ∈ E r ( x);

(b) there is a positive constant c3 for which the inequality F x

ϕλ′(r ( xy−1))dy ≤ c3(r ( x))Qϕλ

′(r ( x)).

Example 3.5.1. Let ϕ(t ) = t α−Q, where Q/λ < α < Q. Then it is easy to see that

ϕ ∈ U λ. Indeed, (a) is obvious due to the properties of the quasi-norm r . Let us show (b).

We have

I :=

F x

ϕλ′(r ( xy−1))dy =

F x

(r ( xy−1))(α−Q)λ′ dy

=

∞0

| B(0,r ( x))| ∩ y ∈ E : (r ( xy−1))(α−Q)λ′ > s|ds

=

r ( x)(α−Q)λ′

0(· · · ) +

∞r ( x)(α−Q)λ′

(· · · ) := I 1 + I 2.

It is clear that I 1 ≤ (r ( x))Qϕλ′(r ( x)), while for I 2 we find that

I 2 ≤ ∞

(r ( x))(α−Q)λ′ s

Q

(α−Q)λ′ ds = c(r ( x))(α−Q)λ′+Q = c(r ( x))Qϕλ′(r ( x)).

50 Alexander Meskhi

The statements of this section are proved just in the same way as for the operator K (see

the previous sections); therefore the proofs are omitted.

Theorem 3.5.1. Let 1 < p ≤ q < ∞ and let k ∈ U p. Then

(i) K is bounded from L p( E ) to Lqv ( E ) if and only if

¯ B := sup j∈Z

¯ B( j) := sup j∈Z

E

2 j+1 \ E 2 j

v( x)ϕq(r ( x))dx1/q

2 jQ/ p′

< ∞;

(ii) K is compact from L p( E ) to Lqv ( E ) if and only if ¯ B < ∞ and lim j→−∞ ¯ B j =

lim j→+∞ ¯ B j = 0.

Theorem 3.5.2. Let 0 < q 1. Suppose that ϕ ∈ U p. Then the


(i) K is bounded from L p( E ) to Lqv ( E );

(ii) K is compact from L p( E ) to Lqv ( E );

(iii)

E

E \ E r ( x)ϕq(r ( y))v( y)dy

p/( p−q)

r ( x)Qp(q−1)/( p−q)dx

( p−q)/( pq)

< ∞.

Let ϕ(t ) = t Qϕ2(t ) and let k ( x) = ϕ(r ( x)). Suppose that

Ku f ( x) = u( x)

E r ( x)

ϕ(r ( xy−1)) f ( y)dy, x ∈ E ,

where u is a measurable function on E .

Theorem 3.5.3. Let 2 ≤ p < ∞ and let ϕ ∈ U p. Then K

u ∈ σ p( L2

( E )) if and only if u ∈ l p( L2

k ( E )). Moreover, there exist positive constants b1 and b2 such that

b1ul p( L2k

( E )) ≤ Kuσ p( L2( E )) ≤ b2ul p( L2k

( E )).


In this chapter we use the material from [7] and [8]. Section 3.4 is published first time.

The two-weight problem for higher-dimensional Hardy-type operators defined on cones

in Rn

involving Oinarov [183] kernels was studied in [234], [97] (see also [221], for Hardy-type transforms on star-shaped regions).

A full characterization of a class of weight pairs (v,w) governing the boundedness of

integral operators with positive kernels from L pw to L

qv , 1 0,

from L p(R+) to Lqv (R+), 1 < p,q < ∞, 1/ p < α < 1 have been obtained in [160] (see also

[198]). This result was generalized in [163] (see also [49], Ch. 2) for integral operatorswith positive kernels involving fractional integrals.

Chapter 4

Potential and Identity Operators

This chapter is devoted to estimates of the measure of non-compactness for potential oper-

ators in weighted Lebesgue spaces defined on Euclidean spaces and homogeneous groups,partial sums of Fourier series, Poisson integrals. The same problem for the identity opera-

tor is also investigated. In some cases we conclude that there is no weight pair for which a

potential operator is compact from one weighted Lebesgue space into another one.

Here keep the notation of Section 1.2.

4.1. Riesz Potentials

Let G be a homogeneous group and let

I α f ( x) = G

f ( y)

r ( xy−1)Q−αdy, 0 < α < Q,

be the Riesz potential operator.

It is well known (see [70], Ch. 6) that I α is bounded from L p(G) to Lq(G), 1 < p,q <∞,

if and only if

q = Qp

Q −α p. (4.1.1)

Moreover, if (4.1.1) holds, then I α is bounded from L pρ p (G) to L

qρq (G) if and only if

sup B

1

| B|

Bρ( x)qdx

1/q 1

| B|

Bρ( x)− p′

dx

1/ p′

< ∞,

where the supremum is taken over all balls B in G (see [171] for Euclidean spaces and [76]

for quasimetric measure spaces with doubling condition).

Our first result in this section is the following statement:

Theorem 4.1.1. Let 1 < p ≤ q < ∞ , 0 < α < Q. Let I α be bounded from L pw(G) to

Lqv (G). Then the following inequality holds

I αK ≥ C α,Q max A1, A2, A3,

52 Alexander Meskhi

where

C α,Q = 1

(2c0)Q−α,

A1 = supa∈G

limr →0

r α−Q

B(a,r )

v( x)dx

1/q

B(a,r )

w1− p′

( x)dx

1/ p′

;

A2 = supa∈G

limr →0

B(a,r )

v( x)dx1/q

( B(a,r ))c

r (ay−1)(α−Q) p′w1− p

′

( y)dy1/ p′

and

A3 = supa∈G

limr →0

B(a,r )

w1− p′

( x)dx1/ p

′ ( B(a,r ))c

r (ay−1)(α−Q)qv( y)dy1/q

(c0 is the constant from the triangle inequality for the homogeneous norms ).The next statement is formulated for the Riesz potentials

J Ω,α f ( x) = Ω

f ( y)| x − y|α−ndy, x ∈Ω,

where Ω is a domain in Rn.

Theorem 4.1.2. Let Ω ⊆ Rn be a domain in Rn. Let 1 < p ≤ q < ∞. If J Ω,α is bounded

from L pw(Ω) to L

qv (Ω), then we have

J Ω,αK ≥ 2α−n B1,

where

B1 = supa∈Ω

limr →0

r α−n

B(a,r )

v( x)dx 1

q

B(a,r )w1− p

′

( x)dx 1

p′.

Further, if Ω = Rn , then

J Ω,αK ≥ 2α−n max B2, B3,

where

B2 = supa∈Rn

limr →0

B(a,r )

v( x)dx1/q

Rn\ B(a,r )

|a − y|(α−n) p′w1− p

′

( y)dy1/ p′

,

B3 = supa∈Rn

limr →0

B(a,r )

w1− p′( x)dx

1/ p′ Rn\ B(a,r )

|a − y|(α−n)qv( y)dy1/q

.

Corollary 4.1.1. Let 1 < p < ∞ , 1 < p < Qα , q = pQ

Q−α p , then there is no weight pair

(v,w) for which I α is compact from L pw(G) to Lq

v (G). Moreover, if I α is bounded from L pw(G)

to Lqv (G) , then

I αK ≥ C α,Q A1,

Potential and Identity Operators 53

where C α,Q and A1 are defined in Theorem 4.1.1.

Proof of Theorem 4.1.1. By Lemma 1.2.7 we have that for a ∈ G and λ > I αK ( L

pw(G), L

qv (G)) there are positive constants β1 and β2, β1 < β2, such that for all τ,s

(τ < β1, s > β2),

B(a,τ)

v( x) I α f ( x)qdx ≤ λq G

f ( x) pw( x)dxq/ p, (4.1.2)

for f ∈ L pw(G), and

B(a,s)c

v( x) I α f ( x)

qdx ≤ λq

B(a,s)

f ( x) pw( x)dx

q/ p

, (4.1.3)

for f with supp f ⊂ B(a,s).

Now assuming f ( x) = χ B(a,r )( x)w1− p

′

( x) in (4.1.2) and observing that B(a,r )

w1− p′

( x)dx < ∞

for all r > 0 (see also [76], Ch. 3 for this fact), we find that B(a,r )

v( x)

B(a,r )

w1− p′( y)

r ( xy−1)Q−αdyq

dx ≤ λq

B(a,r )

w1− p′

( x)dxq/ p

< ∞.

Further, if x, y ∈ B(a,τ), then

r ( xy−1) ≤ c0

r ( xa−1) + r (ay−1)

≤ 2c0τ.

Hence

I αK ≥ C α,Q A1.

If f ( x) = χ B(a,τ)c ( x) w1− p′( x)

r (ay−1)(Q−α)( p′−1), then

B(a,τ)

v( x) B(a,τ)c

w1− p′

( y)dy

r ( xy−1

)Q−α

r (ay−1

)(Q−α)( p

′−1)

q

dx

≤ λq

B(a,τ)c

w1− p′

( x)dx

r (ay−1)(Q−α) p′

q/ p

< ∞.

Let r ( xa−1) < τ and r ( ya−1) > τ. Then

r ( xy−1) ≤ c0

r ( xa−1) + r (ay−1)

≤ c0

τ+ r (ay−1)

≤ 2c0r (ay−1).

Hence, by (4.1.2) we have

1

(2c0)q(Q−α)

B(a,τ)

v( x)dx

B(a,τ)c

w1− p′

( y)dy

r (ay−1)(Q−α) p′

q



54 Alexander Meskhi

≤ λq

B(a,τ)c

w1− p′

( x)dx

r (ay−1)(Q−α) p′

q/ p

.

The latter inequality implies

I αK ≥ 1

(2c0)Q−α A2.

Further, observe that (4.1.3) means that the norm of the operator

¯ I α f ( x) =

B(a,s)

f ( y)dy

r ( y−1a)Q−α

can be estimated as follows:

¯ I α L pw( B(a,s))→ L

qv ( B(a,s)c) ≤ λ.

By the duality arguments we find that

¯ I α L pw( B(a,s))→ L

qv ( B(a,s)c) = ˜ I α

Lq

′

v1−q′ ( B(a,s)c)→ L

p′

w1− p′ ( B(a,s))

,

where

˜ I αg( y) =

B(a,s)c

g( x)dx

r ( xy−1)Q−α.

Indeed, by Fubini’s theorem and Holder’s inequality we have

¯ I α f Lqv ( B(a,s)c) ≤ sup

g L

q′

v ( B(a,s)c)≤1 B(a,s)c g( x)( ¯ I α f ( x))dx

≤ supg

Lq

′

v1−q′ ( B(a,s)c)

≤1

B(a,s)

| f ( y)| ˜ I α(|g|)( y) dy

≤ supg

Lq

′

v1−q′ ( B(a,s)c)

≤1

B(a,s)

| f ( y)| p w( y)dy

1 p

B(a,s)

˜ I α(|g|)

p′

( y)w1− p′

( y)dy

1 p′

≤ ˜ I α B(a,s)

| f ( y)| p w( y)dy 1 p

.

Hence ¯ I α ≤ ˜ I α. Analogously, ˜ I α ≤ ¯ I α.Further, (4.1.3) implies

B(a,s)

w1− p′

( x) ( B(a,s))c

g( y)dy

r ( xy−1)Q−α

p′

dx ≤ λ p′

( B(a,s)c)

|g( x)|q′

v1−q′

( x)dx p

′/q

′

.

Now taking g( x) = χ B(a,s)c ( x)r ( xa−1)(Q−α)(1−q)v( x) in the last inequality we conclude

that I αK ≥ 1

(2c0)Q−α A3.

Theorem 4.1.2 follows in the same manner as Theorem 4.1.1 was obtained. We only

need to use Lemma 1.2.8 instead of Lemma 1.2.7.

4.2. Truncated Potentials

This subsection is devoted to two-sided estimates of the essential norm for the operator

T α f ( x) = B(e,2r ( x))

f ( y)

r ( xy−1

)Q−α

, x ∈ G.

A necessary and sufficient condition guaranteeing the trace inequality for T α defined

on Rn was established in [215]. This result was generalized in [117], [49] (Ch. 6) for the

spaces of homogeneous type. From the latter result (it is also a consequence of Theorem

3.5.1 for E = G) we have

Proposition 4.2.1. Let 1 Q/ p. Then

(i) T α is bounded from L p(G) to Lqv (G) if and only if

B := supt >0

B(t ) := supt >0

r ( x)>t

v( x)r ( x)(α−Q)qdx1/qt Q/ p′ < ∞; (4.2.1)

(i) T α is compact from L p(G) to Lqv (G) if and only if

limt →0

B(t ) = limt →∞

B(t ) = 0.

Theorem 4.2.1. Let 1 < p ≤ q <∞ and let 0 < α< Q. Suppose that T α is bounded from

L pw(G) to L

qv (G). Then the inequality

T αK ( L pw(G)→ L

qv (G)) ≥ C Q,α

lima→0

A(a) + limb→∞

A(b)

holds, where

C Q,α = (2c0)α−Q;

A(a) = sup0<t <a

B(e,a)\ B(e,t )

v( x)r ( x)(α−Q)qdx1/q

B(e,t )

w1− p′( x)dx

1/ p′

;

A(b) = supt >b

B(e,t )c

v( x)r ( x)(α−Q)qdx1/q B(e,t )\ B(e,b)

w1− p

′

( x)dx1/ p′

.

To prove Theorem 4.2.1 we need the following lemma.

Lemma 4.2.1. Let p, q and α satisfy the conditions of Theorem (4.2.1). Then from the

boundedness of T α from L pw(G) to L

qv (G) it follows that w1− p′

is locally integrable on G.

Proof. Let

I (t ) = B(e,t )

w1− p′( x)dx = ∞

56 Alexander Meskhi

for some t > 0. Then there exists g ∈ L p B(e,t )

such that

B(e,t )

gw−1/ p =∞. Let us assume

that f t ( y) = g( y)w−1/ p( y)χ B(e,t )( y). Then we have

T α f t Lqv (G) ≥ χ B(e,t )c T α f t L

qv (G)

≥ c B(e,t )c


B(e,t )

g( y)w−1/ p′( y)dy = ∞.

On the other hand,

f t L pw(G) =

B(e,t )

g p( x)dx < ∞.

Finally we conclude that I (t ) < ∞ for all t , t > 0.

Proof of Theorem 4.2.1. Let λ > T αK ( L p

w(G), Lq

v (G)). Then by Lemma 1.2.7 thereexists a positive constant β such that for all τ1,τ2 satisfying 0 < τ1 < τ2 < β1, and all f with

supp f ⊂ B(e,τ1), the inequality

T α f Lqv ( B(e,τ2)\ B(e,τ1)) ≤ λ f L

pw( B(e,τ1)).

holds. Observe that if r ( x) > τ1 and r ( y) < τ1, then r ( xy−1) ≤ 2c0r ( x). Consequently, taking

f = w1− p′χ B(e,τ1) and using Lemma 4.2.1 we find that

1

(2c0)Q−α B(e,τ2)\ B(e.τ1)

v( x)r ( x)(α−Q)q

dx1q

B(e,τ1)

w1− p′( x)dx

1 p′

≤ λ

for all τ1,τ2,0 < τ1 < τ2 < β1. Hence

1

(2c0)(Q−α)q lim

a→0 A(a) ≤ λ.

Further, by virtue of Lemma 1.2.7 (see (1.2.10)) there exists β2 such that for all s1,s2

with β2 < s1 < s2 the inequality

T α f Lqv ( B(e,s2)c) ≤ λ f L pw( B(e,s2)\ B(e,s1))

holds, where supp f ⊂ B(e,s2)\ B(e,s1). Hence taking f = w1− p′χ B(e,s2)\ B(e,s1) in the previ-

ous inequality and using Lemma 4.2.1 we find that

1

(2c0)Q−α

B(e,s2)c

v( x)

r ( x)(α−Q)q

dx 1

q

B(e,s2)\ B(e,s1)

w1− p′( x)dx

1 p′

≤ λ

which leads us to the estimate

1(2c0)Q−α

limb→∞

A(b) ≤ λ.

Thus we have the desired result.

Theorem 4.2.2. Let 1 < p ≤ q < ∞ and let Q p < α < Q. Suppose that (4.2.1) holds.

Then there is a positive constant C such that

T αK ( L p(G)→ Lqv (G)) ≤ C

lima→0

B(a) + limb→∞

B(b)

,

where

B(a) = supt ≤a

B(e,a)\ B(e,r )


r Q/ p′;

B(b) = supt ≥b

B(e,t )c


r Q − bQ1/ p′

.

Proof. Let 0 < a < b < ∞ and represent T α f as follows:

T α f = χ B(e,a)T α( f χ B(e,a)) +χ B(e,b)\ B(e,a)T α( f χ B(e,b))

+χG\ B((e,b)T α( f χ B(e,b/2c0)) +χG\ B(e,b)T α( f χG\ B(e,b/2c0))

≡:= P1 f + P2 f + P3 f + P4 f ,

where B(e, t ) (t > 0) denotes the closed ball in G with center e and radius t .

For P2, we have

P2 f ( x) = G

k ( x, y) f ( y)dy,

where k ( x, y) = χ B(e,b)\ B(e,a)( x)χ B(e,2r ( x))( y)r ( xy−1)α−Q.

Further observe that G

G

(k ( x, y)) p′dy q

p′

v( x)dx

=

B(e,b)\ B(e,a)

B(e,2r ( x))

(r ( xy−1))(α−Q) p′dy q

p′

v( x)dx

≤ c B(e,b)\ B(e,a)

B(e,r ( x)/2c0)

(r ( xy−1))(α−Q) p′dy

q

p′

v( x)dx.

≤ c

B(e,b)\ B(e,a)

r ( x)(α−Q)q+q/ p′v( x)dx < ∞.

Hence by Lemma 1.3.5 we conclude that P2 is compact for every a and b. Now we

observe that if r ( x) > b and r ( y) < b/2c0, then r ( x) ≤ 2c0r ( xy−1). Further, Lemma 1.3.5

implies that P3 is compact.

Further, repeating the arguments of sufficiency of the proof of Theorem 3.1.1 (see also

proof of Theorem 3.4.1 or [49], Ch. 6) we find that

P1 ≤ C 1 B(a); P4 ≤ C 2 B(b/2c0),

58 Alexander Meskhi

where the constants C 1 and C 2 depend only on p, q, Q and α.Therefore

T α− P2 − P3 ≤ P1 + P4 ≤ c B(a) + B(b/2c0)

.

The last inequality completes the proof.

Theorem 4.2.3. Let p and q satisfy the conditions of Theorem 4.2.2. Suppose that

(4.2.1) holds. Then we have the following two-sided estimate:

c2

lima→0

B(a) + limb→∞

B(b)

≤ T αK ( L p(G), L

qv (G)) ≤ c1

lima→0

B(a) + limb→∞

B(b)

for some positive constants c1 and c2 depending only on Q, α, p, and q.

Theorem 4.2.3 follows immediately from Theorems 4.2.1 and 4.2.2.

4.3. One–sided PotentialsLet

Rα f ( x) =

x 0

f (t )

( x − t )1−αdt , W α f ( x) =

∞ x

f (t )

(t − x)1−αdt ,

where x ∈ R+ and α is a constant satisfying the condition 0 < α < 1.

Theorem 4.3.1. Let 1 RαK ( L pw(R+), L p

v (R+)) and a ∈R+. By Lemma 1.2.8 we have that

a+r a

v( x)( Rα f ( x))qdx ≤ λq a

a−r

f ( x)

pw( x)dx

q p

for small r and non-negative f with supp f ⊂ (a − r ,r ). Hence assuming f ( x) = w1− p′( x)

in the latter inequality we find that

a+r a

v( x)

x 0

w1− p′

( x − t )1−αdt

q

dx ≤ λq a

a−r

w1− p′( x)dx

q p

For x ∈ (a,a + r ) and t ∈ (a − r ,r ), we have that x − t < 2r . Hence

(2r )(α−1)q

a+r

a

v( x)dx

a

a−r

w1− p′(t )dt

q

≤ λq

a

a−r

w1− p′( x)dx

q p

.

Taking into account the boundedness of Rα from L pw(R+) to L

qv (R+) we have w1− p′

∈ Lloc(R+) (see, e.g., [3]). Consequently,

λ≥ (2r )α−1

a+r a

v( x)dx

1q a

a−r

w1− p′( x)dx

1 p′

.

Taking the limit when r → 0 and the supremum over all a ∈R+ in the latter expression,

we have the desired result.

In an analogous manner we can obtain the next statement.

Theorem 4.3.2. Let 1 < p ≤ q < ∞. Suppose that W α is bounded from L pw(R+) to

L pv (R+). Then

W αK ≥ 2α−1 supa∈R+

limτ→0τα−1

a a−τ

v( x)dx

1q a+τ

a

w1− p′( x)dx

1 p′

.

Theorem 4.3.3. Let 1 W αK ( L pw(R+), L

qv (R+)).

Then by Lemma 1.2.8 and the estimate t − x ≤ t − a which holds for x ∈ (a,a + r ) and

t > a + r , the inequalitya+r a

v( x)dx

∞ a+r

f (t )

(t − a)1−αdt

q

≤ λq ∞

a+r

( f ( x)) pw( x)dx

q p

holds, where f ≥ 0, f ∈ L pw(R+), supp f ⊂ (a + r ,∞), a ∈ R+ and r is a small positive

number. Assuming that f (t ) = w1− p′(t )(t − a)( p′−1)(α−1)χ(a+r ,∞)(t ) in the last inequality

and observing that the integral on the right-hand side is finite for this f (see, e.g., [49],

Section 2.2), we have

λ≥ a+r a

v( x)dx 1

q ∞ a+r

w1− p′ ( x)( x − a) p′(1−α)

dx 1

p′

.

Taking the supremum over all a ∈ R+ and passing to the limit when r → 0 in the right-

hand side of the latter inequality, we obtain the desired estimate.

Analogously can be established the following statement for Rα.

Theorem 4.3.4. Let 1 < p < ∞,q < ∞. Suppose that Rα is bounded from L pw(R+) to

L pv (R+). Then

RαK ≥ supa∈R+

limτ→0

a a−r

v( x)dx 1

q a−r −∞

w1− p′( x)

(a − x) p′(1−α)

1 p′

.

60 Alexander Meskhi

4.4. Poisson Integrals

Here we discuss the essential norm of the Poisson integral

P f ( x,t ) =

Rn

f ( y)P( x − y,t )dy, x ∈Rn, t > 0,

where,

P( x,t ) = t (t 2 + | x|2)− n+12 .

In this section we use the following notation:

T B(a,r )

:= ( x, t ) ∈ R

n+1+ ; x ∈ B(a,r ),t < r ; Bn := | B(0,1)| =

2πn2

nΓ ( n2

),

where B(a,r ) ⊂Rn is a ball with center a and radius r and | B(0,1)| is the volume of the ball

B(0,1).Lemma 4.4.1 If ( x, t ) ∈ T ( B) and f ( y) = χ B( y) , then

P f ( x,t ) ≥ Bn

5n+1

2

.

Proof. Let B = B(a,r ). Then we have,

P f ( x, t ) ≥ t

B(a,t )

dy

(t 2 + | x − y|2)n+1

2

≥ t

B(a,t )

dy

(t 2 + |2t |2)n+1

2

≥ Bnt n+1

5n+1

2 t n+1= Bn5− n+1

2 .

In the following statements we keep the notation of Section 1.2.

Lemma 4.4.2. Let w be a weight function on Rn+1+ and let S ∈ F L( L

pw(Rn), Lq(Rn+1

+ )) ,where 1 ≤ p,q <∞. Then for every a ∈Rn and ε> 0, there exist R ∈ F L( L

pw(Rn), Lq(Rn+1

+ ))and positive numbers α,α, 0 < α < α < ∞ , such that for all f ∈ L

pw(Rn) the inequality

(S − R) f Lq(Rn+1+ ) ≤ ε f L pw(Rn)

holds and suppR f ⊂ T ( B(a,α))\T ( B(a,α)).

Proof. It is clear that there exists linearly independent non-negative functions U j ∈ Lq(Rn+1

+ ), j = 1, . . . , N , such that

S f ( x, t ) = N

∑ j=1

β j( f )U j( x,t ),

where β j are bounded linear functionals on L pw(Rn). Further there is a positive constant C

such that N

∑ j=1

|β j( f )| ≤ C f L pw(Rn).

Simple geometric observation shows that we can choose linearly independent Φ j ∈ Lq(Rn+1

+ ) and numbers α j,α j so that

U j −Φ j Lq(Rn+1+ ) < ε/C , j = 1,2, . . . , N ,

and suppΦ j ⊂ T ( B(a,α))\T ( B(a,α)). Let

R f ( x, t ) = N

∑ j=1

β j( f )Φ j( x, t ).

Then we have

S f − R f Lq(Rn+1+ ) ≤

N

∑ j=1

|β j( f )|U j −Φ j Lq(Rn+1+ ) ≤ ε f L

qw(Rn)

for all f ∈ L pw(Rn). Moreover, it is clear that supp R f ⊂ T ( B(a,α))\T ( B(a,α)), where

α = max α j, α = max α j.

The statement below is a slight modification of Lemma 1.2.3; therefore we omit the

proof.

Proposition 4.4.1. Let T be a sublinear and bounded operator from L pw(Rn) to

Lq(Rn+1+ ) , where 1 < p,q < ∞. Then T K = α(T ).

Theorem 4.4.1. Let 1 < p ≤ q < ∞ and let the operator P be bounded from L pw(Rn) to

Lq

v (R

n+1

+ ). Then the following inequality holds:

PK ( L

pw(Rn+1

+ ), Lqv (Rn+1

+ )) ≥ max D1, D2, D3,

where

D1 = Bn

5n+1

2

supa∈Rn

limr →0

T

B(a,r )

v( x, t )dxdt 1

q

B(a,r )

w( x)dx− 1

p

;

D2 = 5−(n+1)/2 supa∈Rn

limr →0

r −n−1

T ( B(a,r ))

t qv( x, t )dxdt

1q

B(a,r )

w( x)1− p′dx

1

p′

;

D3 = supa∈Rn

limr →0

T ( B(a,r ))

v( x, t )dxdt 1

q

( B(a,r ))c

w1− p′( y)dy

(r 2 + 4| y − a|2)

n+1

2

p′

1 p′

.

Proof. Denote Pv f ( x, t ) = v1 p ( x,t )P f ( x, t ). Then

PvK ( L

pw(Rn), Lq(Rn+1

+ )) = PK ( L pw(Rn), Lq

v (Rn+1)).

Let λ > PK ( L

pw(Rn), Lq

v (Rn+1

+ )). Then we see that λ > Pv

K ( L pw(Rn), Lq(Rn+1

+ )). Hence there

exists S ∈ F L( L pw(Rn), Lq(Rn+1

+ )) for which

Pv − S < λ.

62 Alexander Meskhi

Let a ∈ Rn. Then by Lemma 4.4.2 there exist positive numbers α and α and an operator

R ∈ F L( L pw(Rn), Lq(Rn+1

+ ))) such that supp R f ⊂ T ( B(a,α))\T ( B(a,α)) and

S − R ≤ λ− Pv − S

2 .

Hence,Pv − R < λ.

Therefore,

(Pv − R) f Lq(Rn+1+ ) ≤ λ f L

pw(Rn)

for all f ∈ L pw(Rn). Now the latter inequality implies that if r < α, then

T ( B(a,r ))

v( x, t )(Pv f )q( x,t )dxdt ≤ λq Rn

( f ( x)) pw( x)dx q

p

(4.4.1)

for all f ≥ 0, f ∈ L pw(Rn). If we take f ( x) = χ B(a,r )( x) in (4.4.1) and use Lemma 4.4.1 we

find that Bn

5n+1

2

q

T ( B(a,r ))

v( x,t )dxdt ≤ λq

B(a,r )

w( x)dx q

p

which gives the estimate PK ≥ D1.

Assuming that f ( x) = χ B(a,r )( x)w1− p′( x) in (4.4.1) we have

T ( B(a,r ))

t

q

v( x,t ) B(a,r )

w1− p′( y)dy

(t 2 + | x − y|2) n+12 q

dxdt

≤ λq

B(a,r )

w1− p′( x)dx

q p

< ∞.

Further, it is easy to see that for y ∈ B(a,r ) and ( x, t ) ∈ T ( B(a,r )) we have

t 2 + | x − y|2 ≤ r 2 + (2r )2 = 5r 2,

which implies

5−(n+1)/2

r n+1

T ( B(a,r ))

t qv( x, t )dxdt 1

q

B(a,t )

w1− p′( x)dx

1 p′

≤ λ

for all r < α.The latter inequality yields the inequality PK ≥ D2.

To get the estimate PK ≥ D3, we observe that if ( x,t ) ∈ T ( B(a,r )) and | y − a| > r ,

then

t 2 + | x − y|2 ≤ r 2 + 4|a − y|2.

Taking f ( x) = χ x:| x−a|>r ( x)w1− p′( x)r 2 + 4|a − y|2 n+1

2 (1− p′)in (4.4.1), we find that

PvK ≥ D3.

Remark 4.4.1. If w ∈ A p(Rn) (see (1.6.6) ) then it is easy to see that the inequality

PK ≥ D1 of Theorem 4.3.1 can be replaced by

PK ≥ 1

5n+1

2 Bsup

a∈Rn

limr →0

r −n

T ( B(a,r ))

v( x,t )dxdt

1q

B(a,r )

w1− p′( x)dx

1

p′,

where B is defined in (1.6.6).

4.5. Sobolev Embeddings

In this section we deal with the identity operator from a weighted Sobolev space into a

Lebesgue space.

Let Ω ⊆ Rn be a domain and let 1 ≤ p < ∞. Suppose that m is non-negative integer.

Assume that a weight w on R satisfies the condition w ∈ A p(Rn

) (see (1.6.6)). We define theweighted Sobolev space W

m, pw (Ω) as the set of functions u ∈ L

pw(Ω) with weak derivatives

Dαu ∈ L pw(Ω) for |α| ≤ m. Then norm of u in W

m, pw (Ω) is given by

uW m, p

w (Ω) = ∑|α|≤m

Ω

| Dαu( x)| pw( x)dx 1

p

.

It is also defined the space

W

m, pw as the closure of C ∞0 (Ω) in W

m, pw (Ω). Together with

W m, p

w (Ω) we consider the space V m, p

w (Ω) with the norm

uV m, p

w (Ω) = ∑|α|=m

Ω

| Dαu( x)| pw( x)dx 1

p.

For weighted Sobolev inequalities we refer, e.g., to [2], [158],[230], [241].

It is well-known (see, e.g., [230], p. 16) that if w ∈ A p(Rn) then W m, p

w (Ω), W m, p

w (Ω),

V m, p

w (Ω) are Banach spaces.

Now we formulate and prove the main statements of this section.

In the sequel we keep the notation of Section 1.2.

Theorem 4.5.1. Let 1 ≤ p ≤ q < ∞ and let m be any integer such that 0 ≤ m < n.Suppose that w is a weight function on Rn satisfying the condition w ∈ A p(Rn). If W

m, pw (Ω)

is embedded in Lqv (Ω) , i.e., I : W

m, pw (Ω) → L

qv (Ω) is bounded, then

I K ≥ supa∈Ω

limr →0

S m, p(r ,ψ )

−1

B(a,r )

v( x)dx 1

q

B(a,r )

w( x)dx 1

p

,

where S m, p(r ,ψ ) = ∑|α|≤m

r |α| p sup1≤| x|≤2

Dαψ ( x) p 1

p , ψ is a function from C ∞0 (Rn) whose

support is in B(0,2) and equal to 1 in B(0,1).

64 Alexander Meskhi

Proof. Let I vu = vu, then

I vK (W m, p

w (Ω), Lq(Ω)) = I K (W m, p

w (Ω), Lqv (Ω)).

Let λ > α( I v), then there is P ∈ F L(W m, p

w (Ω), Lq(Ω)) such that I v − P < λ. For a ∈ Ω

there exist a positive number α and an operator R ∈ F L(W m, p

w (Ω), Lq(Ω)) such that

P − R ≤ λ− I v − P

2

and supp Ru ⊂Ω\ B(a,α). Hence I v − R ≤ λ. Consequently,

( I v − R)u Lq(Ω) ≤ λuW m, p

w (Ω).

If r < α, then the latter inequality implies

B(a,r )

v( x)|u( x)|qdx 1

q

≤ λuW m, p

w (Ω).

Let us now take ψ ∈ C ∞0 (Rn) which is equal 1 in B(0,1) , supp ψ ⊂ B(0,2) and set

φ = ψ ( x−ar

). Then taking u = ψ in the last inequality we find that B(a,r )

v( x)dx 1

q

≤ λ ∑|α|≤m

Ω

Dαψ ( x)

p

w( x)dx 1

p

≤ λS m, p(r ,ψ ) B(a,r )

w( x)dx 1 p

.

Hence,

I K ≥ 1

S m, p(r ,ψ )

B(a,r )

v( x)dx 1

q

B(a,r )

w( x)dx1 p

for all a ∈Ω and small r .

The next statement follows similarly:

Theorem 4.5.2. Let 1 ≤ p ≤ q < ∞, m ≤ n. Suppose that w ∈ A p(Rn). If V m, p

w (Ω) is

embedded in Lqv (Ω) , i.e., I : V

m, pw (Ω) → L

qv (Ω) is bounded, then

I K (V m, pw (Ω), Lqw(Ω)) ≥ supa∈Rn limr →0S m, p(ψ )−1 B(a,r )

v( x)dx

1q

B(a,r )

w( x)dx 1

p ,

where S m, p(ψ ) = 2m/ p ∑

|α|=m

sup1≤| x|≤2

Dαψ ( x) p 1

p

and ψ is a function from C ∞0 (Rn) with

supp in B(0,2) and value 1 on B(0,1).

Corollary 4.5.1. Let 1 ≤ p < ∞, 0 ≤ m < n p. Suppose that q = np

n−mp, w( x) = | x|β and

v( x) = | x|

βq

p , where β > −npq . Then

I K (V m, p

w (Ω), Lqw(Ω)) ≥

S m, p(ψ )

−1S

1q

− 1 p

n−1

βq p

+ n− 1

p

(β+ n)1q

where S m, p(ψ ) is defined as in the previous statement and S n−1 = 2πn/2

Γ (n/2) .

Example 4.5.2. Let 1 ≤ p ≤ q < ∞ and let 0 ≤ m < n p

. Suppose that V 1, p(Ω) is contin-

uously embedded in Lqv (Ω). Then

I K ≥ supa∈Ω

limr →0

C n, p r m− n

p B(a,r )

v( x)dx 1q,

where C n, p = S − 1

p

n−16e

1+ 31/2

1−31/2

(1−31/2)2 .

This follows from Theorem 4.5.2 taking w ≡ 1 and

ψ ( x) =

1 for | x| < 1,

e1+ 1

(| x|−1)2−1 for 1 ≤ | x| ≤ 2,

0 for | x| > 2.

4.6. Identity Operator

This section is devoted to lower estimates of the measure of non–compactness for the iden-

tity operator I acting from L pw(Ω) to L

qv (Ω), where Ω = [0,π] and q < p.

To prove the main statement we need some lemmas.

Lemma 4.6.1. Let f n( x) = sin 2n x, 0 ≤ x ≤ π. Assume that v ∈ C 1([0,π[) and v,v′ ∈

L1

([0,π[). Thenlim

n,m→∞n=m

π0

v( x)| f n( x) − f m( x)|dx ≥ 1

2

π0

v( x)dx. (4.6.1)

Proof. Let us denote I n,m := π

0 v( x)| f n( x) − f m( x)|dx. We have

I n,m ≥ 1

2

π0

v( x)( f n( x) − f m( x))2dx

= 1

2 π

0

v( x) f 2

n

( x)dx − π

0

v( x) f n( x) f m( x)dx + 1

2 π

0

v( x) f 2

m

( x)dx

:= 1

2 I 1 − I 2 +

1

2 I 3.



66 Alexander Meskhi

For I 1 we have

I 1 = 1

2

π0

v( x)(1 − cos2n+1 x)dx

= 1

2 π

0v( x)dx −

1

2 π

0v( x) cos2n+1 xdx

= 1

2

π0

v( x)dx − 1

2n+2

π0

v′( x) sin2n+1 xdx,

while for I 2 we find that

I 2 = 1

2

π0

v( x)[cos(2n − 2m) x − cos(2n + 2m) x]dx

= 1

2

π0

v( x) cos(2n − 2m) xdx −1

2

π0

v( x) cos(2n + 2m) xdx

:= 12

I 21 − 12

I 22.

It is easy to see that

I 21 = − 1

2n − 2m

π0

v′( x) sin(2n − 2m) xdx;

I 22 = − 1

2n + 2m

π0

v′( x) sin(2n − 2m) xdx.

Hence

I 2 = − 12(2n − 2m)

π0

v′( x) sin(2n − 2m) xdx

+ 1

2(2n + 2m)

π0

v′( x) sin(2n + 2m) xdx

and

I 3 = 1

2

π0

v( x)dx − 1

2m+2

π0

v′( x) sin2n xdx.

Finally, passing n and m to infinity we conclude that

limn,m→∞

n=m

I n,m ≥ 14 π

0v( x)dx + 1

4 π

0v( x)dx = 1

2 π

0v( x)dx.

Note that in the unweighted case the statement that f n contains no subsequence con-

vergent in L1 is made in [143], p. 90.

Lemma 4.6.2. Let f n( x) = sin 2n x, 0 ≤ x ≤ π. Suppose that v is a weight on [0,π]. Then

the inequality (4.6.1) holds.

Proof. Since v is a weight function, we have that v is almost everywhere positive on

[0,π], and v ∈ L1([0,π]). Then there exists a sequence vk , where 0 ≤ vk ( x) ≤ v( x), vk ∈C ∞(0,π) and

π0 v′

k ( x)dx < ∞ such that

limk →∞ π

0

vk ( x)dx = π

0

v( x)dx,

Let us take k so large that

1

2

π0

vk ( x)dx > 1

2

π0

v( x)dx − ε.

By Lemma 4.6.1 we can choose n and m so that π0

vk ( x)| f n( x) − f m( x)|dx ≥ 1

2

π0

vk ( x)dx −ε > 1

2

π0

v( x)dx − 2ε.

From this we conclude that (4.6.1) holds.

Theorem 4.6.1. Let 1 < q < p < ∞ and let Ω = [0,π]. Then there is no pair of weights

(v,w) for which I is compactly embedded from L pw(Ω) to L

qv (Ω). Moreover, if (1.1.1) holds

for some weights v and w on Ω , then

I K ( L pw(Ω), Lq

v (Ω)) ≥ 1

4

π0

v( x)dx

1/q π0

w( x)dx

1/ p

. (4.6.2)

Proof. By Lemma 4.6.2 there exists a sequence f n ⊂ B L∞ ( B L∞ is the closed unit ball

in L∞) such that

( I : L∞→ L1v )( f n) − ( I : L∞→ L1

v )( f m) L1v

= f n − f m L1v> λ−ε,

where λ := 12

π0 v( x)dx and ε is a small positive number. Hence ( I : L∞→ L1

v )( B L∞) cannot

be covered by a finite number of balls of radius λ/2 − ε/2. Thus for entropy numbers of I ,

we have

en( I : ( L∞→ L1v )) ≥ λ/2 − ε/2

for any n ∈ N. Consequently, using the inequality I K ( L∞, L1v ) ≥ β( I ) (see Section 1.2 for

some properties of entropy numbers of bounded linear operators) we find that

I K ( L∞, L1v ) ≥ λ/2 −ε/2.

Further, by Propositions 1.1.3 and 1.1.4 we have

I : L∞w → L pw =

π0

w( x)dx

1/ p

;

I : L pw → Lq

v = π

0 v( x)

w( x) p

p−q

w( x)dx1/q−1/ p

;

I : Lqv → L1

v =

π0

v( x)dx

1/q′

.

68 Alexander Meskhi

Hence

( I : L∞→ L1v ) = ( I : Lq

v → L1v ) ( I : L p

w → Lqv ) ( I : L∞→ L p

w).

From this it follows that

λ−ε

2 ≤ I K ( L∞, L1

v ) ≤ I Lqv → L1 I L∞→ L

pw

I K ( L pw, L

qv ).

Therefore

I K ( L pw, L

qv ) ≥ λ−ε

2 I Lqv → L1

v I L∞→ L

pw

= 1

2(λ− ε)

π0

v( x)dx

−1/q′ π0

w( x)dx

−1/ p

.

But ε

can be taken arbitrarily small. Hence we have (

4.6.2

). Since v

( x

) > 0 almost

everywhere on Ω, we have the desired result.

4.7. Partial Sums of Fourier Series

Here we investigate lower estimates of the essential norm for the partial sums

S n f ( x) = 1

π

π −π

f (t ) Dn( x − t )dt , n ∈ N,

of the Fourier series of f

f ∼ 1

2a0 +∞

∑k =1

(ak cos kx + bk sin kx),

where Dn = 12 +

n

∑k =1

cos kt .

For basic properties of S n see, for instance, [242].

Theorem 4.7.1. Let 1 < p < ∞. Then there is no n ∈ N and weight pair (w,v) on

T := (−π,π) such that S n is compact from L pw(T ) to L p

v (T ). Moreover, if S n is bounded from

L pw(T ) to L

pv (T ) , then

S n ≥ (2 + 21/2)1/2

2π sup

a∈T

limr →0

1

2r

a+r

a−r v( x)dx

1 p 1

2r

a+r

a−r w1− p′

( x)dx 1

p′. (4.7.1)

Proof. Taking λ > S nκ ( L pw(T ), L p

v (T )), by Lemma 1.2.8 we find that

I

v( x)|S n f ( x)| pdx ≤ λ p

I

| f ( x)| pw( x)dx (4.7.2)

for the intervals I := (a − r ,a + r ), where r is a small positive number and supp f ⊂ I .

Let

J 1 =

I

|S n f ( x)| pv( x)dx and J 2 =

I

| f ( x)| pw( x)d ( x).

Suppose that | I | ≤ π4 and n is the greatest integer less than or equal to π

4| I | . Then for x ∈ I

(see [95]),|S n f ( x)| ≥

1

π

I

| f (θ)| sin 3π8

π4n

d θ. (4.7.3)

Using this estimate and taking f := w1− p′( x)χ I ( x) we find that

J 1 ≥1

π sin

3π

8

p

| I |− p

I

v( x)dx

I

w1− p′( x)dx

p

.

On the other hand, due to (4.7.3) it is easy to see that J 2 = I

w1− p′( x)dx < ∞.

Hence, by (4.7.2) we conclude that

λ≥ 1

π sin

3π

8

1

| I |

I

v( x)dx 1

p 1

| I |

I

w1− p′( x)dx

1 p′.

Now passing r to 0, taking the supremum over all a ∈ T and using the fact that sin 3π8

=(2+21/2)1/2

2 we find that (4.7.1) holds.

Corollary 4.7.1. Let 1 < p < ∞ and let n ∈ N. Then

S nκ ( L p(T )) ≥ (2 + 21/2)1/2

2π .

Corollary 4.7.2. Let 1 < p < ∞ and let n ∈ N. Suppose that w( x) = v( x) = | x|α. Then

we have

S nκ ( L pw(T )) ≥

(2 + 21/2)1/2

2π 1

α+ 11 p

1

α(1 − p′) + 11

p′.


Sections 4.1, 4.2 and 4.7 are based on the paper [5]. The results of Section 4.6 are were

derived in the paper [43].

Criteria for the trace inequality ( L p → Lqv boundedness) for the Riesz potentials were

established in [1], [159] (see also the monographs [2], [158], [76] and references therein).

The two-weight problem for the Riesz potentials was solved by E. Sawyer [213], M.

Gabidzashvili and V. Kokilashvili [71], [72] (see also [112]), however, the conditions es-

tablished by M. Gabidzashvili and V. Kokilashvili are more transparent than those of E.

Sawyer.

70 Alexander Meskhi

Necessary and sufficient conditions guaranteeing the compactness of the Reisz poten-

tials from one weighted Lebesgue space into another one have been derived in [47] (see

also [49], Section 5.2).

The two-weight problem for integral transforms with positive kernels defined on quasi-

metric doubling measure spaces were found in [75], [76]. The same problem was solved

in [217], [219] for spaces having a group structure (see also the survey paper [110]). Fortwo–weight inequalities for Poisson integrals we refer to [170], [218] (see also [76], Ch. 3

for integral operators with more general positive kernel).

A full characterization of a class of weight pairs (v,w) governing the boundedness of

one–sided potentials from L pw to L

qv (1 < p < q < ∞) was established in [76], [50] (see also

[49], Ch. 2). We refer also to [153], [154] for the Sawyer–type two-weight criteria for

one–sided potentials.

The one-weight problem for the partial sums of the Fourier series was solved by R. A.

Hunt, Muckenhoupt and R. L. Wheeden [95] (see also the monograph [76]).

Finally we point out that the non–compactness for the majorants of partial sums of theFourier series T f ( x) = supn |S n f ( x)| was investigated in [186].

We are indebted to Professor Peter Bushell for a key idea which led to the proof of

Lemma 4.6.1.

Chapter 5

Generalized One-sided Potentials in

L p( x) Spaces

This chapter deals with boundedness/compactness criteria and measure of non–

compactness for the generalized Riemann-Liouville operator

Rα( x) f ( x) =

x

0 f (t )( x − t )α( x)−1dt , x > 0,

in the L p( x) spaces, where 0 < inf α ≤ supα < 1. In particular, necessary and sufficient

conditions on a weight v guaranteeing the boundedness/ compactness of Rα( x) from L p( x)

to Lq(·)

v are established provided that p satisfies weak Lipschitz condition. When p is an

arbitrary measurable function, we derive sufficient conditions (which are also necessary

for constant exponents) governing the trace inequality for the operator Rα( x). Two-sided

weighted estimates of the measure of non–compactness for Rα( x) are also established.

Throughout this chapter we assume that I is either a bounded interval [0,a] or R+. We

use the notation:

E k := [2k ,2k +1); I k := [2k −1,2k +1), k ∈ Z.

5.1. Boundedness

In this section we establish necessary and sufficient conditions for the boundedness of the

operator Rα( x) from L p( x) to Lq( x)v .

Theorem 5.1.1. Let I = [0,a] be a bounded interval and let 1 < p−( I ) ≤ p( x) ≤ q( x) ≤q+( I ) < ∞. Suppose that (α− 1/ p)−( I ) > 0. Further, assume that p,q ∈ W L( I ). Then the

inequality

vRα( x) f Lq( x)( I ) ≤ c f L p( x)( I ), f ∈ L p(·)( I ) (5.1.1)


Aa := sup0<t <a

Aa(t ) := sup0<t <a

χ(t ,a)( x) v( x) x1−α( x)

Lq( x)( I )

t 1/ p′(0) < ∞.

72 Alexander Meskhi

Moreover, there exist positive constants c1 and c2 such that

c1 Aa ≤ Rα( x) L p( x)( I )→ L

q( x)v ( I )

≤ c2 Aa.

Proof. For simplicity assume that a = 1.

Sufficiency. Suppose that f ≥ 0. We represent Rα( x) f as follows:

( Rα( x) f )( x) =

x/2

0 f (t )( x − t )α( x)−1dt +

x

x/2 f (t )( x − t )α( x)−1dt

:= ( R(1)α( x) f )( x) + ( R

(2)α( x) f )( x).

Hence

vRα( x) f Lq( x)( I ) ≤ vR(1)α( x) f Lq( x)( I ) + vR

(2)α( x) f Lq( x)( I ) := S (1) + S (2).

It is easy to see that if 0 < t < x/2, then x/2 ≤ x − t . Consequently, ( x − t )α( x)−1 ≤ cxα( x)−1,

where the positive constant c does not depend on x. Hence, taking into account Theorem1.4.5 we have

S (1) ≤ c

v( x)

x1−α( x) H f ( x)

Lq( x)( I )

≤ cAa f L p(·)( I ).

Suppose now that g Lq′( x)( I ) ≤ 1. Using Lemmas 1.4.8 and 1.4.9 we find that

1

0v( x)

x

x/2 f (t )( x − t )α( x)−1dt

g( x)dx

≤ c

∑k ∈ Z − E k −1

v( x)χ( x/2, x)

(·) f (·) L

p(·)

( I )

× χ( x/2, x)(·)( x − ·)α( x)−1 L p′(·)( I )g( x)dx

≤ c ∑k ∈ Z −

χ I k −1(·) f (·) L p(·)( I )

E k −1

v( x) xα( x)−1/ p( x)g( x)dx

≤ c ∑k ∈ Z −

χ I k −1(·) f (·) L p(·)( I )

χ E k −1( x)v( x) xα( x)−1/ p( x)

Lq( x)( I )

× χ E k −1(·)g(·)

Lq′(·)( I )

≤ c2

k / p′(0)

∑k ∈ Z −v( x) xα( x)−1

χ E k −1 ( x) Lq( x)( I )χ I k −1 (·) f (·) L p(·)( I )

× χ E k −1(·)g(·)

Lq′(·)( I ) ≤ cAa f L p(·)( I )g Lq′(·)( I ) ≤ cAa f L p(·)( I ).

Taking the supremum with respect to g and applying Lemma 1.4.7, we have the desired

result.

Necessity. Let us take f k ( x) = χ[0,2k −2]( x), where k ∈ Z −. Then by the condition p ∈W L( I ), Lemma 1.4.1 and Proposition 1.4.1 we have

f k L p(·)( I ) ≤ c2k /( p′)+([0,2k +2]) ≤ c2k / p′(0).

On the other hand,

Rα( x) f L

q( x)

vq( x)( I )

≥ cχ E k −1( x)v( x) xα( x)−1 Lq( x)( I ).

Generalized One-sided Potentials 73

Here we used the estimate ( x − t )α( x)−1 ≥ cxα( x)−1 when x ∈ [2k −1,2k ], t < 2k −2. Hence

¯ A := supk ∈ Z −

¯ Ak := supk ∈ Z −

χ E k −1( x)v( x) xα( x)−1 Lq( x)( I )2k / p′(0)

≤ c Rα( x) L p( x)( I )→ L

q( x)v ( I )

.

Let us now take t ∈ I . Then t ∈ [2m−1,2m) for some m ∈ Z −. Consequently,

A1(t ) ≤0

∑k =m

χ E k −1( x)v( x) xα( x)−1 Lq( x)( I )2m/ p′(0) ≤ ¯ A2m/ p′(0)

0

∑k =m

2−k / p′(0) ≤ c ¯ A

≤ c Rα( x) L p( x)( I )→ L

q( x)v ( I )

.

Hence A1 ≤ c Rα( x) L p( x)( I )→ L

q( x)v ( I )

.

Theorem 5.1.2. Let I =R+

and let 1 < p−

( I ) ≤ p( x) ≤ q( x) ≤ q+

( I ) <∞. Suppose that

(α− 1/ p)−( I ) > 0. Further, assume that p,q ∈ W L( I ) and that there is a positive number a

such that q( x) ≡ qc = const , p( x) ≡ pc = const outside [0,a]. Then inequality (5.1.1) holds

if and only if

A∞ := supt >0

A∞(t ) := supt >0

χ(t ,∞)( x) v( x)

x1−α( x)

Lq( x)( I )

t 1/P′(t ) < ∞,

where

P(t ) = p(0), 0 ≤ t ≤ a,

pc, t > a. Moreover, there are positive constants c1 and c2 such that

c1 A∞ ≤ Rα( x) L p( x)( I )→ L

q( x)v ( I )

≤ c2 A∞.

Proof. For simplicity we assume that a = 1. First we prove sufficiency. Suppose that

f ≥ 0. We have

vRα( x) f Lq( x)( I ) ≤ vRα( x) f Lq( x)([0,2]) + vRα( x) f Lq( x)((2,∞)) := I 1 + I 2.

Taking into account Theorem 5.1.1 we find that the condition A∞ < ∞ implies

I 1 ≤ cA∞ f L p( x)([0,2]) ≤ cA∞ f L p( x)( I ).

For I 2, we have

I 2 ≤

v( x)

1

0( x − t )α( x)−1 f (t )dt

Lq( x)((2,∞))

+

v( x) x/2

1( x − t )α( x)−1 f (t )dt

Lq( x)((2,∞))

+v( x) x

x/2( x − t )α( x)−1 f (t )dt

Lq( x)((2,∞))

:= I 2,1 + I 2,2 + I 2,3.

74 Alexander Meskhi

Notice that when t ≤ 1 and x ≥ 2, then ( x − t )α( x)−1 ≤ cxα( x)−1. Consequently, using

Holder’s inequality (see Lemma 1.4.1) we find that

I 2,1 ≤ c

v( x) xα( x)−1

Lq( x)((2,∞)) f χ[0,1] L p(·)( I )χ[0,1] L p(·)( I )

≤ cv( x) xα( x)−1 Lq( x)([1,∞))

f L p(·)( I ) ≤ cA∞ f L p(·)( I ).

It is easy to see that the estimate ( x − t )α( x)−1 ≤ cxα( x)−1 and Theorem 1.3.4 implies

I 2,2 ≤ c

v( x) xα( x)−1

x

1 f (t )dt

Lq( x)([1,∞))

≤ cA∞ f Lq( x)([1,∞)) ≤ cA∞ f Lq( x)( I ),

while Holder’s inequality for the classical Lebesgues spaces yields

( I 2,3)qc ≤ c +∞

∑k =1

E k

v( x)qc x(α( x)−1)qc dx

I k

f pc (t )dt qc/ pc

2k /( pc)′

≤ cAqc∞ f qc

L p(·)( I ).

Necessity follows in the same way as in the case of Theorem 5.1.1. In this case we take the

test functions f t ( x) = χ(t /2,t )( x), t > 0. The details are omitted.

To formulate the next statements we recall that the functions p0( x) and

p0( x) are defined

as follows:

p0( x) := inf y∈[0, x]

p( y); p0( x) := p0( x), 0 ≤ x ≤ a

pc ≡ const, x < a,

where a is a fixed positive number.

Theorem 5.1.3. Let I = [0,a] , where a < ∞. Suppose that p and q are measurable

functions on I and 1 < p−( I ) ≤ p0( x) ≤ q( x) ≤ q+( I ) < ∞. Suppose also that α−( I ) >1/ p−( I ). If

Ba := sup

0<t <a

Ba(t ) := sup

0<t <a a

t

(v( x) xα( x)−1)q( x)t q( x)/( p0)′( x)dx < ∞, (5.1.2)

then inequality (5.1.1) holds.

Proof. For simplicity assume that a = 1. Suppose that S p( f ) ≤ 1, where f ≥ 0. We

have

S q,v( Rα( x)) ≤ 2q−( I )−1

1

0

v( x)

x/2

0( x − y)α( x)−1 f ( y)dy

q( x)

dx

+ 1

0

v( x) x

x/2( x − y)α( x)−1 f ( y)dyq( x)

dx:= 2q−( I )−1( I 1 + I 2).

If 0 < y < x/2, then ( x − y)α( x)−1 ≤ cxα( x)−1, where the positive constant c does not depend

on x. Consequently, using Theorem 1.4.6 we find that

I 1 ≤ c

1

0

v( x)( x/2)α( x)−1

x

0 f ( y)dy

q( x)

dx ≤ C .

By Holder’s inequality with respect to the exponent p0( x) we have

I 2 ≤ 1

0(v( x))q( x)

x

x/2( f ( y)) p0( x)dy

q( x)/ p0( x)

×

x

x/2( x − y)(α( x)−1)( p0)′( x)dy

q( x)/( p0)′( x)

dx.

Now observe that

x

x/2

f ( y) p0( x)dy ≤ [ x/2, x]∩ f ≤1

( f ( y)) p0( x)dy + [ x/2, x]∩ f >1

( f ( y)) p0( x)dy

≤ cx + x

x/2( f ( y)) p( y)dy;

x

x/2( x − y)(α( x)−1)( p0)′( x)dy = c p,α x

(α( x)−1)( p0)′( x)+1.

The latter equality holds because the condition α−( I ) > 1/ p−( I ) guarantees α( x) >

1/ p0( x). Therefore

I 2 ≤ c p,q 1

0

(v( x))q( x) xq( x)α( x)dx

+ 1

0v( x)q( x)

x

x/2 f ( y) p( y)dy

q( x)/ p0( x)

x(α( x)−1)q( x)+q( x)/( p0)′( x)dx

:= c p,q[ I 2,1 + I 2,2].

For I 2,1, we find that

I 2,1 = ∑k ∈ Z −

E k −1

v( x)q( x) x(α( x)−1)q( x) xq( x)dx

≤

∑k ∈ Z −

2kq−( I )/ p+( I ) E k −1

v( x)q( x) x(α( x)−1)q( x)2(k −1)q( x)/( p0)′( x)dx

≤ cB1 ∑k ∈ Z −

2kq−( I )/ p+( I ) ≤ cB1 < ∞,

while taking into account the fact that q( x)/ p0( x) ≥ 1 we have

I 2,2 ≤ c ∑k ∈ Z −

E k −1

(v( x) xα( x)−1)q( x)

x

x/2( f ( y)) p( y)dy

xq( x)/( p0)′( x)dx

≤ c ∑k ∈ Z − E

k −1 v( x) xα( x)−1

q( x)

2(k −1)q( x)/( p0)′( x)dx I k −1

( f ( y)) p( y)dy≤ cB1S p( f ) ≤ cB1 < ∞.

Combining the above-derived estimates we find that I 2 < ∞. The proof follows.

76 Alexander Meskhi

Theorem 5.1.4. Let I = R+. Suppose that p( x) and q( x) are measurable functions on I

and 1 < p−( I ) ≤ p0( x) ≤ q( x) ≤ q+( I ) <∞. Suppose also that α−( I ) > 1/ p−( I ) and there

exists a positive number a such that q( x) ≡ qc = const , p( x) ≡ pc = const outside [0,a]. If

B∞ := sup0<t <∞

B∞(t ) := sup0<t <∞

∞

t

(v( x) xα( x)−1)q( x)t q( x)/(

p0)′( x)dx < ∞, (5.1.3)

then Rα( x) is bounded from L p( x)( I ) to Lq( x)

vq( x) ( I ).

Remark 5.1.1. Notice that (5.1.3) is also necessary for the boundedness of Rα( x) from

L p(R+) to Lqvq (R+), where p and q are constants (see Theorem 5.1.1).

Proof of Theorem 5.1.4. Suppose that f ≥ 0 and S p( f ) ≤ 1. For simplicity assume that

a = 1. We have

S q,v( Rα( x)

f ) = 2

0 v( x)q( x)( Rα f )q( x)( x)dx +

∞

2 v( x)q( x)( Rα( x)

f )qc ( x)dx

:= I 1 + I 2.

Observe that the condition B∞ < ∞ implies Ba < ∞. Consequently, by Theorem 5.1.3 we

conclude that I 1 ≤ c < ∞, while for I 2, we find that

I 2 ≤ c

∞2

(v( x))qc

1

0( x − y)α( x)−1 f ( y)dy

qc

dx

+ ∞

2

(v( x))qc x/2

1

( x − y)α( x)−1 f ( y)dyqc

dx

+

∞2

(v( x))qc

x

x/2( x − y)α( x)−1 f ( y)dy

qc

dx

:= c[ I 2,1 + I 2,2 + I 2,3].

Using Holder’s inequality for Lebesgue spaces with variable exponent (see Lemma

1.4.1) we have

I 2,1 ≤ c

∞

2(v( x))qc

1

0( x − y)α( x)−1 f ( y)dy

qc

dx

≤ c ∞

2(v( x)( x/2)α( x)−1)qc dx

f L p(·)( I )χ[0,1] L p′(·)( I ) ≤ cB∞,

while Theorem 1.3.4 yields

I 2,2 ≤ c

∞1

v( x) xα( x)−1

qc

x

1 f ( y)dy

qc

dx ≤ cB∞.

Now applying the condition α( x) > 1/ p−( I ), we find that

I 2,3 =∞

∑k =1

E k

(v( x))qc

x

x/2 f ( y)( x − y)α( x)−1dy

qc

dx

≤

∞

∑k =1 E k

(v( x))qc x

x/2( f ( y)) pc

dyqc/ pc

×

x

x/2( x − y)(α( x)−1)( pc)′

dy

qc/( pc)′

≤ c∞

∑k =1

E k

(v( x))qc ( x/2)(α( x)−1)qc+qc/( pc)′dx

E k

( f ( y)) p( y)dy

≤ cB∞

∞

∑k =1

E k

( f ( y)) p( y)dy ≤ cB∞ < ∞.

Summarizing the estimates for I 1 and I 2 we have the desired result.

5.2. Compactness

In this section we give the criteria for which the operator Rα( x) is compact from L p(·)( I ) to

Lq(·)v ( I ).

Theorem 5.2.1. Let I = [0,a] , 0 < a <∞ , and let 1 < p−( I ) ≤ p+( I ) ≤ q−( I ) ≤ q+( I ) <∞. Suppose that (α− 1/ p)−( I ) > 0. Further, assume that p,q ∈ W L( I ). Then Rα( x) is

compact from L

p(·)

( I ) to L

q(·)

v ( I ) if and only if (i) Aa < ∞;

(ii) limt →0

Aa(t ) = 0,

where Aa and Aa(t ) are defined in Theorem 5.1.1.

Proof. Sufficiency. For simplicity assume that a = 1 (In this case Aa = A1). We repre-

sent Rα( x) as follows:

Rα( x) f ( x) = R(1)α( x) f ( x) + R

(2)α( x) f ( x),

where

R(2)α( x) f ( x) = χ[0,β]( x) Rα( x) f ( x), R

(1)α( x) f ( x) = χ(β,1]( x) Rα( x) f ( x),

and 0 < β < 1. Observe that by Lemma 1.4.9 we have the following estimates:χ(β,1]( x)v( x)χ[0, x]( y)( x − y)α( x)−1

L p′( y)( I )

Lq( x)( I )

≤χ(β,1]( x)v( x)

χ[0, x/2]( y)( x − y)α( x)−1

L p′( y)( I )

Lq( x)( I )

+χ(β,1]( x)v( x)χ( x/2, x]( y)( x − y)α( x)−1

L p′( y)( I ) Lq( x)

( I

)≤ cχ(β,1]( x)v( x) xα( x)−1/ p( x)

Lq( x)( I )

+cχ(β,1]( x)v( x) xα( x)−1/ p( x)

Lq( x)( I )

< ∞,

78 Alexander Meskhi

because A1 < ∞. Consequently, by Theorem 1.4.8, R(1)α( x) is compact. Further, observe that

the arguments of the proof of Theorem 5.1.1 enable us to conclude that

Rα( x) − R(1)α( x)

L p(·)( I )→ Lq(·)v ( I )

≤ R(2)α( x)

L p(·)( I )→ Lq(·)v ( I )

≤ c sup0<t <β

A1(t ),

where the positive constant c depends only on p, q and α. Passing β to 0 we have that Rα( x)

is compact as a limit of compact operators.

Necessity. Suppose that f t ( x) = t −1/ p(0)χ[0,t /2)( x). Hence, using Holder’s inequality for

L p(·) spaces (see Lemma 1.4.1), Proposition 1.4.1 and the absolutely continuity of the norm

· L p′(·) , we have 1

0 f t ( x)ϕ( x)dx

≤ k ( p) f t (·) L p(·)( I )ϕ(·)χ[0,t /2)(·) L p′(·)( I )

≤ ct −1/ p(0)t 1/ p+([0,t /2])ϕ(·)χ[0,t /2)(·) L p′(·)( I )

≤ cϕ(·)χ[0,t /2)(·) L p′(·)( I ) → 0

as t → 0 for all ϕ ∈ L p′( x)( I ). Hence, f t converges weakly to 0 as t → 0. Further, it is

obvious that

Rα( x) f t Lq(·)v ( I )

≥χ[t ,1)( x)v( x)

t /2

0( x − t )α( x)−1dt

Lq(·)( I )

t −1/ p(0)

≥ ct −1/ p′(0)χ[t ,1)( x)v( x) xα( x)−1

Lq( x)( I ).

Finally we conclude that limt →0 A1(t ) = 0 because the compact operator maps a weakly

convergent sequence into strongly convergent one.

Theorem 5.2.2. Let I = R+ and let 1 < p−( I ) ≤ p( x) ≤ q( x) ≤ q+( I ) < ∞. Suppose

that p( x) ≡ pc = const and q( x) ≡ qc = const when x > a for some positive constant a. Let

(α− 1/ p)−( I ) > 0. Further, assume that p,q ∈ W L( I ). Then Rα( x) is compact from L p(·)( I )

to Lq(·)v ( I ) if and only if

(i) A∞ < ∞;

(ii) limt →0

A∞(t ) = limt →∞

A∞(t ) = 0,

where A∞ and A∞(t ) are defined in Theorem 5.1.2.

Proof. For simplicity assume that a = 1. To prove sufficiency we use the representation

Rα( x) f = ∑5n=1 R

(n)α( x)

f , where

R(1)α( x) f ( x) = χ[0,β)( x)( Rα( x) f )( x),

R(2)α( x) f ( x) = χ[β,γ )( x) Rα( x)(χ[0,β/2] f )( x),

R(3)α( x) f ( x) = χ[β,γ )( x) Rα( x)(χ[β/2,∞) f )( x),

R(4)α( x) f ( x) = χ[γ ,∞) Rα( x)(χ[0,γ /2) f )( x),

R(5)α( x) f ( x) = χ[γ ,∞)( x) Rα( x)(χ[γ /2,∞) f )( x),

where 0 < β < 1/2 < 2 < γ < ∞. Now observe thatχ[β,γ )( x)v( x)

χ[0,β/2)( y)( x − y)α( x)−1

L p′( y)( I )

Lq( x)( I )

≤ cχ[β,γ )( x)v( x) xα( x)−1 L p′( x)( I )χ[0,β/2) L p′(·)( I ) < ∞

because A∞ < ∞. Further,χ[β,γ )( x)v( x)χ[β/2,∞)( y)( x − y)α( x)−1

L p′( y)( I )

Lq( x)( I )

≤

χ[β,γ )( x)v( x)χ[β/2, x/2)( y)( x − y)α( x)−1

L p′( y)( I )

Lq( x)( I )

+χ[β,γ )( x)v( x)χ[ x/2, x

)

( y)( x − y)α( x)−1 L p′( y)( I ) Lq( x)( I )

:= I 1 + I 2.

It is easy to see that Lemma 1.4.1 (for f = χ[β/2,γ /2]) implies

I 1 ≤ c

χ[β,γ )( x)v( x) xα( x)−1

Lq( x)( I )

χ[β/2,γ /2)(·)

L p′(·)( I )< ∞.

Analogously, by Lemma 1.4.9 we can see that I 2 < ∞ because A∞ < ∞. Applying Theorem

5.1.1 we find that

R(1)α( x)

L p(·)( I )→ Lq(·)v ( I )

= Rα( x) L p(·)([0,β))→ L

q(·)v ([0,β))

≤ c sup0<t <β

A∞(t ) → 0

as β → 0. Arguing as in the proof of sufficiency of Theorem 5.1.2 (see also [160] for

constant α), we conclude that the inequality

R(5)α( x) f ( x)

L p( x)([γ ,∞))→ Lq( x)v ([γ ,∞))

≤ c supt >γ /2

∞t

(v( x))qc x(α( x)−1)qc dx

1/qc

(t − γ )1/( pc)′

holds. The latter term tends to 0 when γ → ∞ because limt →∞ A∞(t ) = 0. Finally we con-clude that Rα( x) is compact.

Necessity. The condition A∞ < ∞ is a consequence of Theorem 5.1.2. The condition

limt →0

A∞(t ) = 0 follows in the same manner as in the proof of necessity of Theorem 5.2.1. To

show that limt →∞

A∞(t ) = 0, we can (as above) use the facts that p and q are constants outside

[0,a] and Rα( x) is compact from L p(·)( I ) to Lq(·)v ( I ) if and only if the operator

W α,v f ( x) =

∞ x

v( y) f ( y)( y − x)α( y)−1dy

is compact from Lq′( x)( I ) to L p′( x)( I ).

80 Alexander Meskhi


This section deals with two-sided estimates of the distance between the operator Rα( x) and

the class of compact linear operators from L p(·)( I ) to Lq(·)v ( I ) when I is either finite interval

[0,a] or R+ provided that p and q satisfy the weak Lipschitz condition on I (see Definition

1.4.1).

Let X and Y be Banach spaces. Recall that (see Section 1.4) K ( X ,Y ) (resp. F L( X ,Y ))

denotes the class of compact linear operators (resp. finite rank operators) acting from X to

Y . Let

T K ( X ,Y ) := distT ,K ( X ,Y ); α(T ) := distT ,F L( X ,Y ),

where T is a bounded linear operator from X to Y .

Recall that the symbol P (Ω) denotes the class of those p for which the Hardy-

Littlewood maximal operator M Ω is bounded in L p( x)(Ω) (see section 1.4).

Theorem 5.3.1. Let Ω ⊆ R

n

be a domain. Assume that X is a Banach space. Supposethat 1 < q−(Ω) ≤ q+(Ω) < ∞ and q ∈ P (Ω). Then

T K ( X , Lq(·)(Ω)) = α(T ),

where T is a bounded linear operator from X to Y .

Proof. Let δ > 0. Then there exists an operator K ∈ K ( X , L p(·)(Ω)) such that T −K < T K + δ. By Lemma 1.4.4 there is P ∈ F L( X , L p(·)(Ω)) for which the inequality

K − P < δ holds. This gives

T − P ≤ T − K + K − P ≤ T K + 2δ.

Hence

α(T ) ≤ T K .

The reverse inequality is obvious.

Theorem 5.3.2. Let I = [0,a] , where 0 < a < ∞. Suppose that 1 < p−( I ) ≤ p( x) ≤q( x) ≤ q+( I ) < ∞. Assume that (α− 1/ p)−( I ) > 0. Let p ∈ W L( I ) and let Aa < ∞ (see

Theorem 5.1.1). Then there exists two positive constants b1 and b2 such that

b1A ≤ T K ( L p(·)( I ), L

q(·)v ( I ))

≤ b2A ,

where A := limβ→0

Aβ , Aβ := sup0<t <β

Aa(t ) and Aa(t ) is defined in Theorem 5.1.1.

Theorem 5.3.3. Let I := R+ and let 1 < p−( I ) ≤ p( x) ≤ q( x) ≤ q+( I ) < ∞. Suppose

that (α− 1/ p)−( I ) > 0. Further, assume that p ∈ W L( I ). Suppose also that A∞ < ∞ (see

Theorem 5.1.2). Then there exists two positive constants b1 and b2 such that

b1A ∞ ≤ T K ( L p(·)( I ), L

q(·)v ( I ))

≤ b2A ∞,

where A ∞ := limβ→0

Aβ+ limγ →∞

A(γ ) , Aβ := sup0<t <β

A∞(t ) and A(γ ) := supt >γ

A∞(t ) and A∞(t ) is defined

in Theorem 5.1.2.

Proof of Theorem 5.3.2. For simplicity we assume that a = 1. The upper estimate

follows immediately from the estimate

Rα( x) − R(2)α( x)

L p(·)( I )→ L

q(·)v ( I )

≤ R(1)α( x)

L p(·)( I )→ L

q(·)v ( I )

≤ cAβ,

where R(1)α( x) = χ[0,β]( x) Rα( x) f ( x), R

(2)α( x) = χ[β,1]( x) Rα( x) f ( x), 0 < β < 1 (see the proof of

Theorem 5.2.1 for the details) and the fact that R(2)α( x) is compact (see Theorem 1.4.8). To

get the lower estimate we take a positive number λ so that λ > T K ( L p(·)( I ), L

q(·)v ( I ))

. Notice

that

Rα( x)K

L p(·)( I ), L

q(·)v ( I )

= Rα( x),vK ( L p(·)( I ), Lq(·)( I )),

where

Rα( x),v f ( x) = v( x) x

0 f (t )( x − t )α( x)−1dt .

Consequently, by Theorem 5.3.1 λ > α( Rα( x),v).

Hence, there exist g1, . . . ,g N ∈ Lq(·)( I ) such that

α( Rα( x),v) ≤ Rα( x),v − F < λ,

where

F f ( x) = N

∑ j=1

α j( f )g j( x),

α j are linear bounded functionals in L p(·)( I ) and gi are linearly independent. Further, there

exist g1, . . . , g N such that support of gi is in [σi,a], 0 < σi < a, and

Rα( x),v − F 0 < λ,

where F 0 f ( x) = ∑ N j=1α j( f )g j( x). Suppose that σ = minσ j. Then suppF 0 f ⊂ [σ,a]. Let

0 < t < β < σ and let f be a non-negative function with support in [0, t /2] such that

f L p(·)( I ) ≤ 1. Consequently, for such an f we have

λ ≥ λ f L p(·)( I ) ≥ χ[0,β]( x)( Rα( x),v f ( x) − F 0 f ( x)) L p( x)( I )

≥ χ[t ,β]( x)( Rα( x),v f )( x) Lq( x)( I )

≥χ[t ,β]( x)v( x)

t /2

0( x − y)α( x)−1 f ( y)dy

Lq( x)( I )

≥ c

χ[t ,β]( x)v( x) xα( x)−1

Lq( x)( I )

t /2

0 f ( y)dy

.

Taking the supremum with respect to f we find thatχ[t ,β]( x)v( x) xα( x)−1

Lq( x)( I )

χ[0,t /2](·) L p′(·)( I ) ≥ cAβ.

Taking into account the condition p ∈ W L( I ) and Lemma 1.4.1 we have the desired result.

In a similar way can be proved Theorem 5.3.3; therefore we omit it.

82 Alexander Meskhi


The statements of Sections 5.1 and 5.2 were presented in [9].

Necessary and sufficient conditions on a weight function v guaranteeing the bounded-

ness of the Riemann-Liouville operator Rα from L p(R+) to Lqv (R+), where α, p and q are

constants were established in [160] for α > 1/ p and in [118] for 0 < α < 1/ p (see also[163], [49], Ch. 2). Later the same problems were investigated independently in [198],

[199].

For weighted inequalities for the classical integral operators in variable exponent func-

tion spaces we refer to the papers [122]–[133], [57], [52], [136], [31], [53], [33], [51],

[119], [105], [106], [210], [211], [90], etc (see also the surveys [111], [208] and references

therein).

Integral–type necessary conditions and sufficient conditions governing the compactness

of the Hardy operator H from L p(·)( I ) to Lq(·)v ( I ) were established in [52]. We refer also

to [57] for the compactness of the potential-type operators in weighted L p(·)

spaces withspecial weights. A dominated compactness theorem in L

p(·)ρ (Ω, µ), where µ(Ω) < ∞ and

ρ is a power-type weight was established in [200]. This result was applied to fractional

integral operators over bounded sets.

In [163] (see also [49], Ch. 2) two-sided estimates of the measure of non–compactness

for one-sided potentials acting from the classical Lebesgue space into the classical weighted

Lebesgue space were obtained. Lower and upper estimates of the measure of non–

compactness for the Hardy operator in variable exponent Lebesgue spaces were studied

in [52].

Chapter 6

Singular Integrals

In this chapter the essential norm for singular integrals (Hilbert transforms, Cauchy in-

tegrals, Riesz transforms, Caldeon-Zygmund operators), generally speaking, in weightedLebesgue spaces with non–standard growth is estimated from below.

We keep the notation of Chapter 1.

6.1. Hilbert Transforms

Suppose that H is the Hilbert transform (see Section 1.6 for the definition).

The following statements give the lower estimate of the essential norm for H in classical

weighted Lebesgue spaces.

Theorem 6.1.1. Let 1 < p <∞. Suppose that H is bounded from L pw(R) to L

pv (R). Then

H K ( L pw(R)→ L

qv (R)) ≥ sup

a∈Rlimr →0

a+r

av( x)dx

1/ p ∞a+r

w1− p′( x)

( x − a) p′ dx

1/ p′

.

Theorem 6.1.2. Let 1 < p < ∞ and let w ∈ A p(R). Then

H K ( L pw(R)) ≥

1

2 max ¯ A1, ¯ A2,

where

¯ A1 = supa∈R

limr →0

1

r

a+r

aw( x)dx

1/ p1

r

a

a−r w1− p′

( x)dx

1/ p′

;

¯ A2 = supa∈R

limr →0

1

r

a

a−r w( x)dx

1/ p1

r

a+r

aw1− p′

( x)dx

1/ p′

.

Proof of Theorem 6.1.1. Let a ∈R and λ> H K ( L pw(R)→ L

pv (R)). By Lemma 1.2.8 there

exists a positive number β such that for all r < β and all f with the support in ⊂ (a + r ,∞)we

have a+r

av( x)| H f ( x)| pdx ≤ λ p

∞a+r

w( x)| f ( x)| pdx. (6.1.1)

84 Alexander Meskhi

It is obvious that if y ∈ (a + r ,∞) and x ∈ (a,a + r ), then y − x ≤ y − a. If we assume that

f ( y) = w1− p′( y)χ(a+r ,∞)( y)( y − a)1− p′

in (6.1.1), then we find that

λ p ∞a+r

w1− p′ ( x)( x − a)− p′ dx ≥ a+

r

av( x)| H f ( x)| pdx

≥

a+r

av( x)dx

∞a+r

w1− p′( x)

( x − a) p′ dx

p

.

Now if we show that

J (a,r ) := ∞

a+r w1− p′

( x)( x − a)− p′dx < ∞

for all a ∈ R and r < β, then we are done because (6.1.1) implies the inequality a+r

av( x)dx

1/ p ∞a+r

w1− p′( x)

( x − a) p′ dx

1/ p′

≤ λ

for all a ∈ R and r < α.Suppose the opposite: there exists a ∈ R and r > 0 such that J (a,r ) = ∞. By duality

arguments there exists a function g ∈ L p(a + r ,∞) such that g ≥ 0 and ∞a+r

g( x)w−1/ p( x)

x − adx = ∞.

Further, we take the function

φ( x) = g( x)χ(a+r ,∞)( x)w−1/ p( x)

in the two-weight inequality

H φ L pv ≤ cφ L

pw

and, consequently, we conclude that

∞ =

a+r

av( x)dx

∞

a+r

g( x)w−1/ p( x)

x − adx

p

≤ c

∞

a+r (g( x)) pdx < ∞

which is impossible unless v( x) = 0 almost everywhere on (a,a + r ).

Proof of Theorem 6.1.2. First notice that by Theorem 1.6.1 the condition w ∈ A p(R)implies the boundedness of H in L

pw(R). Let λ > H K ( L

pw(R) and let a ∈ R. Using again

Lemma 1.2.8 we have that there is β> 0 such that if 0 < r < β and supp f ⊂ (a − r ,a), then a+r

aw( x)| H f ( x)| pdx ≤ λ p

a

a−r w( x)| f ( x)| pdx. (6.1.2)

It is clear that x − y < 2r when y ∈ (a − r ,a), x ∈ (a,a + r ). Let us put f ( y) =

w1− p′

( y)χ(a−r ,a)( y) in (6.1.2). Then we observe that

λ p a

a−r w1− p′

≥ a+r

aw( x)| H f ( x)| pdx ≥

1

(2r ) p

a+r

aw( x)dx

a

a−r w1− p′

( x)dx

p

Singular Integrals 85

from which, taking into account the fact that w1− p′∈ Lloc(R) (see e.g. [95]), it follows

12

¯ A1 ≤ λ. Analogously we have 12

¯ A2 ≤ λ.

Corollary 6.1.1. Let p = 2 , w( x) = | x|α, −1 < α < 1. Then

H K ( L2w(R)) ≥

1

1 +α .

This follows immediately from Theorem 6.1.2 if we assume that v( x) ≡ w( x) ≡ | x|α and

a = 0.

Corollary 6.1.2. Let 1 < p < ∞ and let w ∈ A p(R). Then

H K ( L pw(R)→ L

pw(R))) ≥

1

2(1 + 2 H L pw(R))

supa∈R

limr →0

A(r ,a) p (R),

where A(r ,a

) p (R) is defined in Definition 1.6.1 , and

H L pw(R) := H L

pw(R)→ L

pw(R).

Proof. First note that the condition w ∈ A p(R) and Theorem 1.6.1 imply the inequali-

ties: a+r

aw( x)dx ≤ 2 p H p

L pw(R)

a

a−r w( x)dx, (6.1.3)

a

a−r w( x)dx ≤ 2 p H p

L pw(R)

a+r

aw( x)dx. (6.1.4)

Indeed, if we put f ( y) = χ(a−r ,a)( y) in the one-weight inequality R

w( x)| H f ( x)| pdx ≤ H p

L pw(R)

R

w( x)| f ( x)| pdx, (6.1.5)

then we find that

Rw( x)| H f ( x)| pdx ≥ a+r

a

w( x) a

a−r

dy

x − y p

dx ≥ 1

2 p

a+r

a

w( x)dx.

On the other hand, f p

L pw(R)

= a

a−r w < ∞. Hence (6.1.3) holds.

Analogously we can show that (6.1.4) holds. Let us introduce the notation:

W (b,c) :=

c

bw( x)dx

1/ p

; V (b,c) :=

c

bw1− p′

( x)dx

1/ p′

.

Further, due to Theorem 6.1.2 and (6.1.3) − (6.1.4) we conclude that

1

2r W (a − r ,a + r )V (a − r ,a + r ) ≤

1

2r W (a − r ,a)V (a − r ,a)

+ 1

2r W (a − r ,a)V (a,a + r ) +

1

2r W (a,a + r )V (a − r ,a)

86 Alexander Meskhi

+ 1

2r W (a,a + r )V (a,a + r ) ≤

2 H L pw(R)

2r W (a,a + r )V (a − r ,a)

+ 1

2r W (a − r ,a)V (a,a + r ) +

1

2r W (a,a + r )V (a − r ,a)

+2

H

L

p

w(R)2r

W (a − r ,a)V (a,a + r ) ≤ 2(1 + 2 H L pw(R)) H K ( L

pw(R))

when r is small.

6.2. Cauchy Singular Integrals

Let Γ be a smooth Jordan curve and let S Γ be the Cauchy singular integral operator along Γ

(see Section 1.6 for the definition).

We begin with the following Lemma:

Lemma 6.2.1. Let 1 < p < ∞. Suppose that S Γ is bounded from L pw(0, l) to L p

v (0, l).

Then

S I :=

I w1− p′

(s)ds < ∞, for all subintervals I of (0, l).

Proof. Let S I = ∞ for some I . Consequently, it follows that there exists g ∈ L p( I ), g ≥0, such that

I

g(t )w−1/ p(t )dt = ∞. Now let φ(s) = f (t (s)) = g(s)w−1/ p(s)χ I (s). Let I =

(a − r ,a) ⊂ (0, l). Without loss of generality we can assume that I ′ = (a,a + r ) ⊂ (0, l). We

have (see [109], [107], p. 56)

|S Γ f (t (σ))| ≥ 1

2π

I

φ(s)

s −σds ≥

1

4πr

I

φ(s)ds (6.2.2)

for σ ∈ I ′. Consequently,

|S Γ f (t (σ))| ≥ 1

4πr

I

φ(s)dsχ I ′ (σ) (6.2.3)

for any σ ∈ (0, l). Hence, using inequality (6.2.3), we find that

S Γ f L pv (0,l) ≥

1

4πr

I

φ(s)ds

χ I ′ L pv (0,l)

= 1

4πr

I

g(s)w−1/ p(s)ds

=∞

χ I ′ L

pv (0,l) = ∞.

On the other hand,

φ L pw(0,l) = g L p( I ) < ∞.

This contradicts the boundedness of S Γ from L pw(0, l) to L

pv (0, l).

Theorem 6.2.1. Let 1 < p < ∞. Suppose that Γ is a Jordan smooth curve. Then there

exists no weight pair (v,w) such that the operator S Γ is compact from L pw(0, l) to L

pv (0, l).

Moreover, if S Γ is bounded from L pw(0, l) to L

pv (0, l) , then the inequality

S Γ K ( L pw(0,l)) ≥

1

4π

max A1, A2

holds, where

A1 := supa∈(0,l)

limr →0

1

r

a+r

av(s)ds

1/ p1

r

a

a−r w1− p′

(s)ds

1/ p′

;

A2 := supa∈(0,l)

limr →0

1

r

a+r

aw(s)ds

1/ p1

r

a

a−r w1− p′

(s)ds

1/ p′

.

Proof. Let S Γ be bounded from L pw(0, l) to L

pv (0, l), λ > S Γ K ( L

pw(0,l), L p

v (0,l)) and a ∈

(0, l). Then, using Lemma 1.2.8 there exists a positive number β such that for all r < β we

have

S Γ f Lqv ( I (a,r )) ≤ λ f L

pw(0,l), f ∈ L p

w(0, l), (6.2.4)

where I (a,r ) = (a − r ,a + r ).

Let I 1 := (a − r ,a), I 2 := (a,a + r )ϕ(s) = f (t (s)) ≥ 0

) and suppϕ⊂ I 2. Then we have

the estimate similar to (6.2.2):

|S Γ f (t (σ))| ≥

1

2π I 2

ϕ(s)

s −σds

χ I 1 (σ) ≥

1

4r π I 2

ϕ(s) ds

χ I 1 (σ),

Analogously,

|S Γ f (t (σ))| ≥

1

4r π

I 1

ϕ(s)ds

χ I 2 (σ).

By Lemma 6.2.1 we have that w1− p′is locally integrable. Let ϕ(s) = w1− p′

(s)χ I 1 (s). Then

by (6.2.4) we have

1

(4π) p

1

r

a+r

aw1− p′

(s)ds

p−11

r

a

a−r v(s)ds

≤ λ p, a ∈ (0, l).

The latter inequality implies 1

4π A2 ≤ λ.

Analogously, it follows that1

4π A1 ≤ λ.

This completes the proof.

Theorems 6.2.1 and 1.6.2 imply the next statement:

Theorem 6.2.2. Let 1 < p < ∞. Suppose that Bl < ∞ , where Bl is defined in Theorem

1.6.2. Then the inequality

S Γ K ( L pw(0,l)) ≥

1

4π max A1, A2

88 Alexander Meskhi

holds, where

A1 := sup

a∈(0,l)

limr →0

1

r

a+r

aw(s)ds

1/ p1

r

a

a−r w1− p′

(s)ds

1/ p′

;

A2 := supa∈(0,l)

limr →0

1

r

a+r

aw(s)ds

1/ p1

r

a

a−r w1− p′

(s)ds1/ p′

.

The next two corollaries follow in the same manner as in the case of the Hilbert trans-

form.

Corollary 6.2.1. Let p = 2 , w( x) = xα , where −1 < α < 1. Then

S Γ K ( L pw(0,l)) ≥

1

4π(1 −α2)1/2.

Corollary 6.2.2. Let 1 < p < ∞ and let w ∈ A p(0, l) (see Theorem 1.6.2). Then

S Γ K ( L pw(0,l)) ≥

1

4π(4πS Γ L pw(0,l) + 1)

supa∈(0,l)

limr →0

A(r ,a) p ,

where

A(r ,a) p =

1

2r

a+r

a−r w(s)ds

1/ p 1

2r

a+r

a−r w1− p′

(s)ds

1/ p′

and S Γ L pw(0,l) is the norm of S Γ in L pw(0, l).

6.3. Riesz Transforms

Let R j f , 1 ≤ j ≤ n, be the Riesz transforms of f defined by (1.6.5).

Theorem 6.3.1. Let 1 < p < ∞. Then there are no pair of weights (v,w) and integer j,

1 ≤ j ≤ n, such that the operator R j is compact from L pw(Rn) to L

pv (Rn). Moreover, if R j is

bounded from L pw(Rn) to L

pv (Rn) for some j, then the following inequality holds

R jK ≥ An esssupa∈Rn

v(a)

w(a)

1/ p

,

where An = γ n Bn

2n+1n3/2 , Bn = πn/2

Γ (1+n/2).

Proof. Let R j be bounded from L pw(Rn) to L

pv (Rn) for some 1 ≤ j ≤ n. By Lemma

1.6.2 we have that w1− p′∈ Lloc(R). Using Lemma 1.2.8, for λ > R j

K

L

pw(Rn), L p

w(Rn) and

a ∈ Rn, there exists a positive number β such that for all 0 < τ < β and f ∈ L pw(Rn) the

inequality

R j,v f L p( B(a,r )) ≤ λ f L pw(Rn) (6.3.1)

holds, where R j,v f = vR j f .




Let a = (a1, . . . ,an) and let

E j,a :=

x = ( x1, . . . , xn) ∈ Rn : max| xi − ai|, 1 ≤ i ≤ n = x j − a

∩ B(a,τ).

It is obvious that

B(a,τ) = ∪n

j=1 E j,a ∩ (− E j,a).Let ¯ f = f χ− E j,a and let x ∈ E j,a. Then

| R j ¯ f ( x)| = γ n

− E j,a

f (t ) x j − t j

| x − t |n+1dt

≥ γ n

− E j,a

f (t ) ( x j − t j)

| x − t |nn1/2( x j − t j)dt = γ n

n1/2

− E j,a

f (t )

| x − t |ndt

≥ γ n

n1/2(2τ)n − E j,a

f (t )dt ,

where f ≥ 0.

Further, using the latter estimates and assuming f = w1− p′in (6.3.1) we have

cn

E j,a

v( x)dx

− E j,a

w1− p′( x)dx

p

≤ λ p

− E j,a

w1− p′( x)dx,

where

cn = γ

pn

(n1/22n) pτnp.

On the other hand, notice that

1

| E j,a|

E j

v( x)dx → v(a) a.e..

Indeed, we have

| E j,a| = | B(a,τ)|

2n=

Bnτn

2n. (6.3.2)

Therefore

1

| E j,a|

E j,a

v( x) − v(a)

≤ 1

| E j,a|

E j,a

|v( x) − v(a)|dx

≤ 2n

| B(a,τ)|

B(a,τ)|v( x) − v(a)|dx −→ 0 a.e.

as τ→ 0.

Analogously, 1

| − E j,a|

− E j,a

|w1− p′( x) − w1− p′

(a)|dx

→ 0

when τ→ 0.

Hence

cn| E j,a|

p v(a)

w(a) ≤ λ

p

for almost every a ∈ Rn, which on the other hand, together with (6.3.2) implies the desired

estimate.

90 Alexander Meskhi

6.4. Calderon–Zygmund Operators

In this section we discuss the essential norm of the Calderon-Zygmund singular integral

operator K (see (1.6.1) for the definition of K ).

Our aim in this section is to prove the following statement:

Theorem 6.4.1. Let 1 < p < ∞. Suppose that conditions (1.6.2)-(1.6.4) are satisfied.

Then there exists no weight pair (v,w) such that the singular integral operator K is com-

pact from L pw(Rn) to L

pv (Rn). Moreover, if K is bounded from L

pw(Rn) to L

pv (Rn) , then the

inequality

K K ≥ c esssupa∈Rn

v(a)

w(a)

1/ p

(6.4.1)

holds, where the positive constant c depends only on n, b and t (see (1.6.4) and Lemma

1.6.1 for b and t ).

Proof. Let K be bounded from L pw(Rn) to L

pv (Rn). Lemma 1.6.2 implies that w1− p′

is locally integrable. Further, repeating the arguments of Theorem 6.3.1 we see that

by Lemma 1.2.7 for λ > K K ( L pw(Rn), L p

w(Rn)) and a ∈ Rn, there exists β > 0 and R ∈

F L

L

pw(Rn), L p

v (Rn)

with supp R ⊂ Rn \ B(a,β) for all f ∈ L

pw(Rn) such that for all f ∈

L pw(Rn) the inequality

K f L p(Rn) ≤ λ f L pw(Rn) (6.4.2)

holds.

Let B := B(a,r ), where r < β. Suppose that B′ is the translation of B in the direction of

u, i.e. B′ = B(a + ru,r ), where u = tu0, t is taken so that the conditions of Lemma 1.6.1 aresatisfied and u0 is the unit vector chosen so that (1.6.4) holds. Let f be any non-negative

function supported in B. Consider T f ( x) for x ∈ B′. We have

K f ( x) =

B

k ( x − y) f ( y)dy

with x = a + ru + rx′, | x′| < 1. Since y ∈ B, we find that y = a + ry′ for | y′| < 1. Thus

x − y = r (u + r ( y′ − x′)) = r (u + v) with |v| < 2. Further, Lemma 1.6.1 and condition (1.6.4)

yield

|K f ( x)| ≥ 12

f B|k (ru)| ≥ c f B 1| B| , (6.4.3)

for all x ∈ B′, where | B| denotes a measure of B and c is the positive constant depending

only on n, b and t . Due to inequality (6.4.2) we obtain B′

v( x)

B

k ( x − y) f ( y)dy

pdx ≤ λ p

B( f ( y)) pw( y)dy

for all non-negative f with supp f ⊂ B. Let f ( x) = w1− p′( x)χ B( x). Then using (6.4.3), we

find thatc p

| B| p

B′v( x)dx

f

p B ≤ λ p

B

w1− p′( y)dy.




Consequently by Lemma 1.6.2 we have

c pv B′ ((w1− p′) B) p−1 ≤ λ p. (6.4.4)

Further, observe that the equality

limr →0 v B′ = v(a) (6.4.5)

holds for almost all a. This follows from the obvious fact

|v B′ − v(a)| ≤ c 1

| B|

B |v( x) − v(a)|dx → 0

as r → 0, where B = B(a,r (t + 1)) and c is a positive constant.

Inequalities (6.4.4) and (6.4.5) yield

c v(a)w(a)1/ p

≤ λ

for almost every a (here the positive constant c depends only on a, n and t ). As λ is an

arbitrary number greater than K K , we conclude that (6.4.1) holds.

6.5. Hilbert Transforms in L p( x) Spaces

Here we estimate from below the essential norm of the Hilbert transform acting between

two weighted Lebesgue spaces with variable exponent. In particular, we show that there is

no weight pair (v,w) and a function p ∈ W L(R) for which H is compact from L p(·)w (R) to

L p(·)v (R).

First we formulate the main results of this section

Theorem 6.5.1. Let p ∈ P (R) and let H be bounded from L p(·)w (R) to L

p(·)v (R). Then

the following estimate holds

H K ≥ (1/2) max A1, A2, (6.5.1)

where

A1 = supa∈R

limr →0

1

r χ(a−r ,a)v L p(·)(R)χ(a,a+r )w−1

L p′(·)(R),

A2 = supa∈R

limr →0

1

r χ(a,a+r )v L p(·)(R)χ(a−r ,a)w−1 L p′(·)(R).

Theorem 6.5.2. Let p ∈ P (R). Suppose that H is bounded from L p(·)w (R) to L

p(·)v (R).

Then

H K ≥ max B1, B2, (6.5.2)

where

B1 = supa∈R

limr →0

χ(a,a+r )(·)v(·) L p(·)(R)χ(a+r ,+∞)(·)w−1(·)(· − a)−1 L p′(·)(R);

92 Alexander Meskhi

B2 = supa∈R

limr →0

χ(a−r ,a)(·)v(·) L p(·)(R)χ(−∞,a−r )(·)w−1(·)(a − ·)−1 L p′(·)(R).

The next statement also holds.

Theorem 6.5.3. Let p ∈ P (R). Suppose that H is bounded from L p(·)w (R) to L

p(·)v (R).

Then H K ≥ (1/4) maxC 1,C 2,

where

C 1 = supa∈R

limr →0

χ I (a,r )(·)v(·) L p(·)(R)χR\ I (a,r )(·)w−1(·)| · −a|−1 L p′(·)(R);

C 2 = supa∈R

limr →∞

χR\ I (a,r )(·)v(·)| · −a|−1 L p(·)(R)χ I (a,r )(·)w−1(·) L p′(·)(R).

Now we give another estimate of the essential norm of H .

Theorem 6.5.4. Let p ∈ P (R). Assume that H is bounded from L p(·)w (R) to L

p(·)v (R).

Then

H K ≥ (1/4) supa∈R

limr →0

χ(a−r ,a+r )v L p(·)(R)w−1(·)(r + |a − ·|)−1 L p′(·)(R).

Corollary 6.5.1. Let p satisfy (1.4.1) and (1.4.2). Then there is no weight pair (v,w)

such that H is compact from L p(·)w (R) to L

p(·)v (R). Moreover, if H is bounded from L p(·)(R)

to L p(·)(R) , then the inequality

H K

L

p(·)w (R), L

p(·)v (R)

≥ e− A/( p−)2

4 sup

a>0

limr →0

1

2r

a+r

a−r (v(t )) p(t )dt

1/ p−( I (a,r ))

×

1

2r

a+r

a−r (w(t ))− p′(t )dt

1/( p′)−( I (a,r ))

> 0

holds.

Corollary 6.5.2. Let p satisfy conditions (

1.4.1

) and

(1.4.2

). Then

H K

L p(·)(R)

) ≥ (1/4)e− A/( p−)2

,

where the positive constant A is from (1.4.1).

Remark 6.5.1. It is known that if

v( x) =

| x|−1/ p ln−1 e

| x| , if 0 < x ≤ 1,

1, if x > 1;

w( x) = | x|−1/ p, if 0 < x ≤ 1,1, if x > 1

,

where p is a constant with 1 < p < ∞, then H is bounded from L pw(R) to L

pv (R) (see [46]).

Based on this fact and Theorem 6.5.2 we have the following estimate:

H K

L

pw(R), L p

v (R)

≥ ( p − 1)−1/ p.

To prove the main results of this section, we need some lemmas.

Lemma 6.5.1. Let T be a linear map from L p(·)w to L

p(·)v . Then T is bounded (resp.

compact ) if and only if T v,w is bounded (resp. compact ) in L p(·) , where T v,w f := vT ( f w−1).

Moreover, T L

p(·)w (R)→ L

p(·)v (R)

= T v,w L p(·)(R)→ L p(·)(R). Further, if T is bounded, then

T K

L

p(·)w (R), L

p(·)v (R)

= T v,wK ( L p(·)(R)).

Proof. The first part of the lemma can be checked immediately. For the second part

observe that

T v,w − P L p(·)(R)→ L p(·)(R) = T − Pv,w L

p(·)w (R)→ L

p(·)v (R)

,

where Pv,w f = 1/vP( f w).

Lemma 6.5.2. Let H be bounded from L p(·)w (R) to L

p(·)v (R). Then

G I (·) L p′(·)(R) < ∞,

for all bounded intervals I, where G I ( x) = w−1( x)(| I |/2 + | x − a I |)−1 and a I is the center

of I.

Proof. Suppose that G I /∈ L p′(·)(R) for some interval I := (a I − τ,a I + τ). By Lemma

1.4.6 we have that there exists g ∈ L p′(·)(R) such that g ≥ 0 and R

G I ( x)g( x)dx = ∞.

Hence either S I := +∞

a I G I g =∞ or

a I

−∞G I g =∞. Suppose that S I =∞. Then we take f ( x) =g( x)χ(a I ,+∞)( x). Then using Lemma 6.5.1 we find that

∞ > χ(a I ,+∞)(·)g(·) L p(·)(R) ≥ H

−1

L p(·)w → L p(·)v H v,w f L p(·)(R)

≥ H −1

L p(·)w → L

p(·)v

χ(a I −τ,a I )(·)v(·) L p(·)(R)

+∞ a I

g(t )G I (t )dt = ∞.

In the last inequality we used the inequality t − x ≤ (t − a I ) +τ which is true for all x, t with

x ∈ (a I − τ,a I ) and t > a I .

Proposition 6.5.1. Let H be bounded from L p(·)w (R) to L

p(·)v (R). Then

sup I

χ I v L p(·)(R)w−1(·)(| I |/2 + |a I − ·|)−1 L p′(·)(R)

≤ 4 H L

p(·)w (R)→ L

p(·)v (R)

, (6.5.3)

where I is a bounded interval and a I is the center of I .

94 Alexander Meskhi

Proof. Due to Lemma 6.5.2 we have that G I ∈ L p′(·)(R), where

G I ( x) = w−1( x)(| I |/2 + | x − a I |)−1.

Let g ≥ 0 and let g ∈ L p(·)(R). Then by Holder’s inequality for L p(·) spaces (see Lemma

1.4.1) we see that J I :=

R

g(t )G I (t )dt < ∞. (6.5.4)

Let us choose r ∈R so that

+∞ r

g(t )G I (t )dt = (1/2) J I . (6.5.5)

Now observe that if x ∈ I (−∞,r ) and t ∈ (r ,+∞), then 0 < t − x ≤ |t − a I | + |a I − x| <

|t − a I | + | I |/2. Hence for such an x we have (recall that H v,w f = vH ( f /w))

H v,wg( x) ≥ v( x)

+∞ r

g(t )G I (t )dt = ( J I /2)v( x). (6.5.6)

Due to Lemma 6.5.1 we have

g L p(·)(R) ≥ H −1

L p(·)w → L

p(·)v

H v,wg L p(·)(R) ≥ ( J I /2) H −1χ I

(−∞,r )(·)v(·) L p(·)(R).

In a similar manner we can find that

g L p(·)(R) ≥ ( J I /2) H −1χ I

(r ,+∞)(·)v(·) L p(·)(R).

Now taking the supremum with respect to g with g L p(·)(R) ≤ 1 and using Lemma 1.4.7 we

conclude that (6.5.3) holds.

Proof of Theorem 6.5.1. By Lemma 6.5.1 we have

H v,wK ( L p(·)) = H K ( L

p(·)w , L

p(·)v )

.

Let λ> H K ( L

p(·)w , L

p(·)v )

. Then by the previous equality and Theorem 5.3.1 we have that

λ > α( H v,w). Hence there exists P ∈ F L( L p(·)) such that

H v,w − P < λ.

Let us take an arbitrary a ∈ R. By Lemma 1.4.5 there exist a positive number β and R ∈F L( L p(·)) such that

R − P < λ− H v,w − P

2

and

supp R f ⊂R\ I (a,β)

for all f ∈ L p(·), where I (a,β) = (a −β,a +β). Consequently,

( H v,w − R) f L p(·) ≤ λ f L p(·), f ∈ L p(·)(R),

where supp R f ⊂R\ I (a,β). From the latter inequality it follows that if 0 < τ < β, then

χ I (a,τ) H v,w f L p(·)(R) ≤ λ f L p(·)(R), f ∈ L p(·)(R). (6.5.7)

Let g be a non-negative function such that g ∈ L p(·)(R). By Lemma 6.5.2 we have that

w−1(·)χ I (a,τ)(·) L p(·)(R) < ∞. Hence

I (a,τ) gw−1 < ∞. Now observe that for t ∈ (a,a + τ)and x ∈ (a − τ,a), 0 < t − x < 2τ. Consequently, assuming f = gχ(a,a+τ) in (6.5.7) we find

that

∞ > λgχ(a,a+τ) L p(·)(R) ≥ χ(a−τ,a)(·) H v,w f (·) L p(·)(R)

≥ 1

2τχ(a−τ,a)(·)v(·) L p(·)(R) a+τ

agw−1.

Taking the supremum with respect to all g with g L p(·)(R) ≤ 1, applying Lemma 1.4.7

and passing to the limit as τ→ 0, we have that

H K ≥ (1/2) A1.

In a similar manner we can show that

H K ≥ (1/2) A2.

Proof of Theorem 6.5.2. Using the arguments from the proof of Theorem 6.5.1, for

λ > H K and a ∈ R we have that inequality (6.5.7) holds. Let us take f = gχ(a+τ,+∞) in

(6.5.7), where g is non-negative and g L p(·)(R) ≤ 1. Due to Lemma 6.5.2 we have

χ(a+τ,+∞)(·)w−1(·)(· − a)−1 L p′(·)(R) < ∞.

This implies

+∞

a+τ

g(t )w−1(t )(t − a)−1dt < ∞.

Further,

∞ > λgχ(a,a+τ) L p(·)(R) ≥ χ(a,a+τ)(·) H v,w f (·) L p(·)(R)

≥ χ(a,a+τ)(·)v(·) L p(·)(R)

+∞

a+τg(t )(t − a)−1w−1(t )dt

.

Taking the supremum with respect to all such a g we conclude that

H K ≥ B1.

Analogously, H K ≥ B2.

96 Alexander Meskhi

Proof of Theorem 6.5.3. Repeating the arguments of Theorem 6.5.1 we arrive at in-

equality (6.5.7). Further, assume that supp f ⊂ (a + τ,+∞) in (6.5.7), where f ≥ 0 and

f L p(·)(R) ≤ 1. Then we observe that

∞ > λ f χ(a+τ,a) L p(·)(R) ≥ χ I (a,r )(·) H v,w f (·) L p(·)(R)

≥ (1/2)χ I (a,τ)(·)v(·) L p(·)(R)

+∞

a+τ f (t )(t − a)−1w−1(t )dt

.

In the latter inequality we used the estimate t − x ≤ 2(t − a) which holds for all t and x

with t > a + τ, | x − a| < τ.Consequently, taking the supremum over all such an f we conclude that

λ≥ (1/2)χ I (a,τ)(·)v(·) L p(·)(R)χ(a+τ,+∞)(·)w−1(·)(· − a)−1 L p′(·)(R).

Arguing in the same manner as above we shall see that

λ≥ (1/2)χ I (a,τ)(·)v(·) L p(·)(R)χ(−∞,a−τ)(·)w−1(·)(a − ·)−1 L p′(·)(R).

Summarazing the estimates derived above, we conclude that

χ I (a,τ)(·)v(·) L p(·)(R)χR\ I (a,r )(·)w−1(·)| · −a|−1 L p′(·)(R)

≤ χ I (a,τ)(·)v(·) L p(·)(R)χ(−∞,a−r )(·)w−1(·)| · −a|−1 L p′(·)(R)

+χ I (a,τ)(·)v(·) L p(·)(R)χ(a+r ,+∞)(·)w−1(·)| · −a|−1 L p′(·)(R) ≤ 4λ.

These estimates lead us to the conclusion H K ≥ (1/4)C 1.Further, notice that due to Theorem 5.3.1 and Lemma 1.4.5 we have that there exists a

sufficiently large positive number γ and R ∈ F L( L pw) such that

H v,w f − R f L p(·)(R) ≤ λ f L p(·)(R), f ∈ L p(·)(R),

where λ > H K

L

p(·)w , L

p(·)v

, and supp R f ⊂ I (a,γ ). Consequently,

χR\ I (a,s) H v,w f L p(·)v (R) ≤ λ f L p(·)(R), f ∈ L p(·)(R),

for all s, s > γ . Let f be a non-negative function and let supp f ⊂ I (a,s). Then

∞ > λ f χ I (a,s) L p(·)(R) ≥ χ(a+s,+∞)(·) H v,w f (·) L p(·)(R)

≥ (1/2)χ I (a+s,+∞)(·)v(·) L p(·)(R)

a+s

a−s f (t )w−1(t )dt

.

Taking the supremum with respect to f with f L p(·)(R) ≤ 1 we find that

λ≥ (1/2)χ(a+s,+∞)(·)(· − a)−1v(·) L p(·)(R)χ I (a,s)(·)w−1(·) L p′(·)(R).

Analogously,

λ≥ (1/2)χ(−∞,a−s)(·)(a − ·)−1v(·) L p(·)(R)χ I (a,s)(·)w−1(·) L p′(·)(R).

Consequently, H K ≥ (1/4)C 2.

Proof of Theorem 6.5.4. Let a ∈ R and let τ be so small positive number that (6.5.7)holds. Let us denote I := (a −τ,a +τ). We repeat the arguments from the proof of Proposi-

tion 6.5.1. Let g be a non-negative function such that g L p(·)(R) ≤ 1. According to Lemma

6.5.2 we have that (6.5.4) holds. Now we choose r , r ∈ R, so that (6.5.5) is fulfilled. Let

I ∩ (−∞,r ) = /0. Observe that 0 < (t − x) ≤ (t − a) + τ whenever x ∈ I ∩ (−∞,r ) and t > r .

Using the arguments similar to those of Theorem 6.5.1 we have that (6.5.7) holds. Substi-

tuting f = gχ(r ,+∞) in (6.5.7) we find that

∞ > λχ(r ,+∞)g L p(·)(R) ≥ ( J I /2)χ I ∩(−∞,r )(·)v(·) L p(·)(R),

where J I is defined by (6.5.4).Analogously, if I ∩ (r ,+∞) = /0, then

∞ > λχ(−∞,r )g L p(·)(R) ≥ ( J I /2)χ I ∩(r ,+∞)(·)v(·) L p(·)(R).

Summarazing these inequalities and taking the supremum with respect to g and a, and

passing to the limit as τ→ 0 we have the desired result.

Proof of Corollary 6.5.1. Let a ∈ R. Suppose that v(a) > 0 and w(a) < ∞. Due to the

condition p ∈ W L(R), Proposition 1.4.1, Theorems 6.5.4 and Remark 1.4.1 we have

H K L p(·)w (R), L p(·)

v (R)≥ (1/4)lim

r →0(1/2r )χ I (a,r )(·)v(·) L p(·)(R)w−1(·)χ I (a,r )(·)

L p′(·)(R)

≥ (1/4)limr →0

(2r )−1/ p+( I (a,r ))−1/( p+( I (a,r )))′

a+r

a−r (v(t )) p(t )dt

1/ p−( I (a,r ))

×

a+r

a−r (w(t ))− p′(t )dt

1/( p′)−( I (a,r ))

≥ e− A/( p

−)2

4 lim

r →0 1

2r a+r

a−r (v(t )) p(t )dt 1/ p−( I (a,r ))

×

1

2r

a+r

a−r (w(t ))− p′(t )dt

1/( p′)−( I (a,r ))

> 0.

Proof of Corollary 6.5.2. Let I := (a − r ,a + r ). Applying the condition p ∈ W L(R),

Theorems 6.5.4, Proposition 1.4.1 and Remark 1.4.1 we have

H K ( L p(·)) ≥ (1/4) sup

a∈Rlimr →0

1

2r χ I (·) L p(·)(R)(·)χ I (·)

L p′(·)(R)

≥ e− A/( p−)2

4 sup

a∈Rlimr →0

1

2r (2r )1/ p+( I )(2r )1/( p+( I ))′

= e− A/( p−)2

4 .

98 Alexander Meskhi

6.6. Cauchy Singular Integrals in L p( x) Spaces

Here we discuss lower estimates of the essential norm for the Cauchy singular integral

operator S Γ along a smooth Jordan curve Γ on which the arc length is chosen as a parameter.

We begin with the following Lemma:Lemma 6.6.1. Let 1 < p− ≤ p( x) ≤ p+ < ∞. If S Γ is bounded from L

p(·)w (0, l) to

L p(·)v (0, l) , then

S I := w−1χ I L p′(·)(0,l) < ∞,

for all subintervals I of (0, l).

Proof. Let S I = ∞ for some I . This implies that there exists some g ∈ L p(·)( I ), g ≥ 0,such that

I

g(t )w−1(t )dt = ∞. Let

φ(s) = f (t (s)) = g(s)w−1(s)χ I (s)

and let I = (a − r ,a) ⊂ (0, l). We can assume that I ′ = (a,a + r ) ⊂ (0, l). We have (see

[107], [109])

|S Γ f (t (σ))| ≥ 1

2π

I

φ(s)

s −σds ≥

1

4πr

I

φ(s)ds

for σ ∈ I ′ and sufficiently small r . Thus

|S Γ f (t (σ))| ≥ 1

4πr I

φ(s)dsχ I ′ (σ) (6.6.1)

for any σ.Hence using inequality (6.6.1) we find that

v(σ)(S Γ f )(σ) L p(σ)(0,l) ≥χ I ′ (σ)v(σ)

1

2π

I

φ(s)

s −σds

L p(σ)(0,l)

≥ 1

4πr I

φ(s)dsχ I

′ (σ)v(σ) L

p(σ)

(0,l)

= 1

4πr

I

g(s)w−1(s)ds

=∞

χ I ′ (σ)v(σ) L p(σ)(0,l) = ∞.

On the other hand,

w(·)φ(·) L p(·)(0,l) = χ I (·)g(·) L p(·)(0,l) < ∞

Now the inequalityv(S Γ f ) L p(·)(0,l) ≤ cw(·)φ(·) L p(·)(0,l)

implies the desired result.

Theorem 6.6.1. Let 1 < p− ≤ p( x) ≤ p+ < ∞ and let S Γ be bounded from L p(·)w (0, l) to

L p(·)v (0, l). Then

S Γ K ( L

p(·)w (0,l), L

p(·)v (0,l))

≥ 1

4π max ˜ A1, ˜ A2,

where

˜ A1 = supa∈(0,l)

limr →0

1r

χ(a−r ,a)v L p(·)(0,l)χ(a,a+r )w−1 L p′(·)(0,l)

and

˜ A2 = supa∈(0,l)

limr →0

1

r χ(a,a+r )v L p(·)(0,l)χ(a−r ,a)w−1

L p′(·)(0,l).

Proof. Let

λ > S Γ K ( L

p(·)w (0,l), L

p(·)v (0,l))

.

Then using the fact that

S Γ K ( L

p(·)w (0,l), L

p(·)v (0,l))

= S Γ ,v,wK ( L p(·)(0,l)),

where

S Γ ,v,w f

(t (s)) = v(s)

S Γ f w−1

(t (s)) and the equality (see Lemma 5.3.1)

S Γ ,v,wK ( L p(·)(0,l)) = α(S Γ ,v,w),

we have

λ > α(S Γ ,v,w).

Let us take an arbitrary a ∈ (0, l). By Lemma 1.4.5 there exists a positive number β and an

operator R ∈ F L( L p(·)(0, l)) such that

R − P < λ− S Γ ,v,w − P

2

and supp R f ⊂ (0, l)\ I (a,β). Consequently,

(S Γ ,v,w − R) f (t (·)) L p(·)(0,l) ≤ λφ L p(·)(0,l)

for all f ∈ L p(·)(0, l). If we choose r so small that 0 < r < β then the inequality above leads

us to the estimate

χ I (a,r )S Γ ,v,w f (t (·)) L p(·)(0,l) ≤ λφ L p(·)(0,l) (6.6.2)

which holds for all φ ∈ L p(·)(0, l). According to Lemma 6.6.1, w−1χ I L p′(·)(0,l) < ∞ for all

subintervals I ⊂ (0, l). Let g(s) ≥ 0, g ∈ L p(·)(0, l). Then by Holder’s inequality for L p(·)

spaces we find that I (a,r )

gw−1 ≤ cgχ I L p(·)(0,l)w−1χ I L p′(·)(0,l) < ∞.

Further, if s ∈ I 2 = (a,a + r ) and σ ∈ I 1 = (a − r ,a), then using (6.6.1) we haveS Γ ,v,w f (t (σ))= v(σ)

S Γ ( f w−1)(t (σ))≥

v(σ)

4πr

I 2

w−1(s)φ(s)ds. (6.6.3)

100 Alexander Meskhi

Taking φ(s) = g(s)χ I 2 (s) in (6.6.2) and taking into account (6.6.3) we see that

∞ > λg(s)χ I 2 (s) L p(·)(0,l) ≥ χ I 1 (·)S Γ ,v,w f (·) L p(·)(0,l)

≥ χ I 1 v L p(·)(0,l) I 2

w−1(s)φ(s)ds.

Taking the supremum over all such a g we get

λ≥ 1

4πr v(·)χ(a−r ,a)(·) L p(·)(0,l)w−1(·)χ(a,a+r )(·)

L p′(·)(0,l).

This inequality implies

S Γ K ( L

p(·)w (0,l), L

p(·)v (0,l))

≥ 1

4π˜ A1.

Let us now take σ ∈ I 2 and let φ(s) = f (t (s)) be nonnegative function with suppφ ⊂ I 1.

Then S Γ f (t (σ))≥

1

2π

I 1

φ(s)

s −σds ≥

1

4πr

I 1

φ(s)ds.

Thus, for σ ∈ I 2 and sufficiently small r ,S Γ f (t (σ))≥

1

4πr

I 1

φ(s)dsχ I 2 (σ).

Taking φ(s) = g(s)χ I 1 (s) in (6.6.2) we get

∞ > g(s)χ I 1 (s) L p(·)(0,l) ≥ 14πr

v(·)χ I 2 (·) L p(·)(0,l) I 1

w−1(s)g(s)ds.

If we take the supremum with respect to g and use the fact that

w−1 L p′(·)( I 1) ≤ supg

L p(·)(0,l)≤1

l

0χ I 1 (t )g(t )w−1(t )dt

,we obtain

λ≥ 1

4πr χ(a,a+r )v L p(·)(0,l)χ(a−r ,a)w−1

L p′(·)(0,l).

Taking the supremum over a ∈ (0, l) and passing to the limit when r → 0, we conclude that

S Γ K ( L

p(·)w (0,l), L

p(·)v (0,l))

≥ 1

4π˜ A2.

Theorem 6.6.2. Let 1 < p− ≤ p( x) ≤ p+ < ∞ and let S Γ be bounded in L p(·)w (Γ ). Then

S Γ K ( L

p(·)w (0,l))

≥ 1

4π(4πS Γ + 1) supa∈(0,l)

limr →0

A(r ,a) p(·)

,

where A

(r ,a) p(·) =

1

2r χ(a−r ,a+r )(·)w(·) L p(·)(0,l)χ(a−r ,a+r )(·)w−1(·)

L p′(·)(0,l)

and S Γ is the operator norm.




Proof. Let f (t (s)) = χ I 1 (s), where I 1 = (a − r ,a). Suppose that I 2 := (a,a + r ). We

have

J := S Γ f L

p(·)w (0,l)

≥ 1

2π

χ I 2 (·)

I 1

f (t (s))

· − sds

L p(·)w (0,l)

≥ 1

4πχ I 2 (·)

L p(·)w (0,l)

.

Also,

J ≤ S Γ χ I 1 (·) L

p(·)w (0,l)

.

Combining these inequalities we obtain

χ I 2 (·)w(·) L p(·)(0,l) ≤ 4πS Γ χ I 1 (·)w(·) L p(·)(0,l). (6.6.4)

Analogously,

χ I 1 (·)w(·) L p(·)(0,l) ≤ 4πS Γ χ I 2 (·)w(·) L p(·)(0,l). (6.6.5)

Now applying (6.6.4) and (6.6.5) we find that

A(r ,a) p(·) =

1

2r χ(a−r ,a+r )(·)w(·) L p(·)(0,l)χ(a−r ,a+r )(·)w−1(·)

L p′(·)(0,l)

≤ 1

2r

χ I 1 (·)w(·) L p(·)(0,l) + χ I 2 (·)w(·) L p(·)(0,l)

×

χ I 1 (·)w−1(·) L p′(·)(0,l) + χ I 2 (·)w−1(·)

L p′(·)(0,l)

=

1

2r χ I 1

(·)w(·) L

p(·)

(0,l)χ

I 1(·)w−1(·)

L p′(·)

(0,l)

+χ I 1 (·)w(·) L p(·)(0,l)χ I 2 (·)w−1(·) L p′(·)(0,l)

+χ I 2 (·)w(·) L p(·)(0,l)χ I 1 (·)w−1(·) L p′(·)(0,l)

+χ I 2 (·)w(·) L p(·)(0,l)χ I 2 (·)w−1(·) L p′(·)(0,l)

≤

1

2

4πS Γ

1

r χ I 2 (·)w(·) L p(·)(0,l)χ I 1 (·)w−1(·)

L p′(·)(0,l)

+

1

r χ I 1 (·)w(·) L p(·)(0,l)χ I 2 (·)w−1

(·) L p′(·)(0,l)

+1

r χ I 2 (·)w(·) L p(·)(0,l)χ I 1 (·)w−1(·)

L p′(·)(0,l)

+4πS Γ 1

r χ I 1 (·)w(·) L p(·)(0,l)χ I 2 (·)w−1(·)

L p′(·)(0,l)

.

Using Theorem 6.6.1 for v ≡ w, taking the supremum over all a ∈ (0, l) and passing to

the limit as r → 0 and we find that

supa∈(0,l)

limr →0

A(r ,a)

p(·) ≤

1

216π2

S Γ

S Γ K ( L

p(·)w (0,l))

+ 4π

S Γ K ( L

p(·)w (0,l))

4πS Γ K ( L

p(·)w (0,l))

+ 16π2S Γ S Γ K ( L

p(·)w (0,l))

102 Alexander Meskhi

= 4πS Γ K ( L

p(·)w (0,l))

+ 16π2S Γ S Γ K ( L

p(·)w (0,l)))

.

Therefore

supa∈(0,l)

limr →0

A(r ,a) p(·) ≤ 4πS Γ

K ( L p(·)w (0,l))

4πS Γ + 1

.

Finally we have the desired result.


This chapter is based on the papers [165], [166], [43], [6].

For the estimates of the essential norm S Γ K ( L pw(Γ )), where Γ is a Lyapunov curve and

w is a power-type weight, see [141], [142]. In [68] it was shown that when w ∈ A2(Γ ), then

S T K ( L pw(T )) = 1 if and only if log w ∈ V MO(T ), where T is the unit circle.

It should be pointed out that in the one-weight case the lower estimates of the essential

norm of S Γ in Banach function spaces, where Γ is a Carlesson curve, have been derived in[102], [103]. In particular, these results give the lower estimates of S Γ K ( L p

w), 1 < p < ∞,

where w is the Muckenhoupt weight.

The one-weight problem for the Hilbert transform and Caldeon-Zygmund singular in-

tegrals was solved in [95], [24] (see also the monographs [73], [224], [76] and references

therein).

For two-weight inequalities for the Hilbert transform and singular integrals on Rn in

Lebesgue spaces we refer to the papers [172], [46], [190], [23], [178], [179], [201], [28],

[147] (see also the monographs [76], [49], [233] and references therein). We notice that

the conditions of [178] and [147] on weight pairs involve the operator itself. The sameproblems for singular integrals defined on nilpotent groups were studied in [113], [114],

[86] (see also [49], [76], [87] and references therein). It should be emphasized that the

two-weight problem for the Hilbert transform remains still open.

For weighted inequalities for the operator S Γ in classical Lebesgue spaces we refer to

[109], [107], [76], [49].

Weighted estimates for S Γ in L p(·) spaces were obtained in [120], [121], [127]–[130],

[132], [134].



References

[1] D. R. Adams, Trace inequality for generalized potentials. Studia Math. 48 (1973),

99–105.

[2] D. R. Adams and L. I. Hedberg, Function spaces and potential theory, Springer–

Verlag, Berlin. 1996.

[3] K. Andersen and E. Sawyer, Weighted norm inequalities for the Riemann-Liouville

and Weyl fractional integral operators. Trans. Amer. Math. Soc. 308(1988), No. 2,

547–558.

[4] T. Ando, On the compactness of integral operators. Indag. Math. (N.S.) 24 (1962),

235–239.

[5] M. Asif and A. Meskhi, On a measure of non–compactness for maximal and potential

operators. J. Inequal. Appl., 2008, Art. ID 697407, 19 pages.

[6] M. Asif and A. Meskhi, On an essential norm for the Hilbert transforms in L p( x)

spaces. Georgian Math. J. 15 (2008), No. 2, 208–223.

[7] U. Ashraf , M. Asif and A. Meskhi, Boundedness and compactness of posi-

tive integral operators on cones of homogeneous groups. Positivity (Online: DOI

10.1007/s11117-008-2217-8).

[8] U. Ashraf, M. Asif and A. Meskhi, On a class of bounded and compact higher di-

mensional kernel operators, Proc. A. Razmadze Math. Inst. 144 (2007), 121–125.

[9] U. Ashraf, V. Kokilashvili and A. Meskhi, Weight characterization of the trace in-

equality for the generalized Riemann-Liouville transform in L p( x) spaces. Math. In-

equal. Appl. (to appear).

[10] R. Banuelos and P. Janakiraman, L p-bounds for the Beurling–Ahlfors transform.

Trans. Amer. Math. Soc. 360 (2008), No. 7, 3603–3612.

[11] J. Bergh and J. Lofstrom, Interpolation spaces: An introduction. Grundlehren Math.

Wiss., Vol. 223, Springer–Verlag, Berlin, 1976.

[12] C. Bennett and R. Sharpley, Interpolation of operators. Pure and Applied Mathemat-ics. 129, Academic Press, Inc., Boston, MA, 1988.



104 References

[13] M. Sh. Birman and M. Solomyak, Estimates for the singular numbers of integral

operators (Russian). Uspekhi Mat. Nauk. 32 (1977), No.1 , 17–82; English transl.

Russian Math. Surveys 32 (1977).

[14] M. Sh. Birman and M. Solomyak, Asymptotic behaviour of the spectrum differential

operators with anisotropically homogeneous symbols. Vestnik Leningrad. Univ. (Rus-sian) 13 (1977), 13–21, English Transl. Vestnik Leningrad Univ. Math. 10 (1982).

[15] M. Sh. Birman and M. Solomyak, Quantitative analysis in Sobolev embedding theo-

rems and applications to spectral theory. AMS Transl. Series 2, Vol. 114, 1980.

[16] M. Sh. Birman and M. Solomyak, Estimates for the number of negative eigenvalues

of the Schrodinger operator and its generalizations. Adv. Soviet Math. 7( 1991), 1–

55.

[17] A. B¨ottcher and Y. I. Karlovich,

Carleson Curves, Muckenhoupt weights, and

Toeplitz operators. Progress in Mathematics , Vol. 154, Birkhauser, 1997.

[18] J. S. Bradley, Hardy inequality with mixed norms. Canad. Math. Bull. 21 (1978),

405–408.

[19] S. M. Buckley, Estimates for operator norms on weighted spaces and reverse Jensen

inequalities. Trans. Amer. Math. Soc. 340 (1993), No. 1, 253–272.

[20] B. Carl, Entropy numbers of diagonal operators with applications to eigenvalue prob-

lems. J. Approx. Theory 32(1981), 135–150.

[21] A. Carleson, A., An interpolation problem for bounded analytic functions. Amer. J.

Math. 80 (1958), 921–930.

[22] C. Capone, D. Cruz-Uribe SFO, A. Fiorenza, The fractional maximal operator on

variable L p spaces. Revista Mat. Iberoamericana 3(23) (2007), 747–770.

[23] A. Cianchi, A note on two-weight inequalities for maximal functions and singular

integrals. Bull. London Math. Soc. 29 (1997), No. 1, 53–59.

[24] R. R. Coifman and Ch. Fefferman, Weighted norm inequalities for maximal functionsand singular integrals. Studia Math. 51 (1974), 241–250.

[25] J. Creutzig and W. Linde, Entropy numbers of certain summation operators. Geor-

gian Math. J. 8 (2001), No. 2, 245–274.

[26] D. Cruz-Uribe, SFO, A. Fiorenza, J. M. Martell and C. Perez, The boundedness of

classical operators on variable L p spaces. Ann. Acad. Sci. Fenn. Math. 31 (2006),

239–264.

[27] D. Cruz-Uribe, A. Fiorenza and C. J. Neugebauer, The maximal function on variable

L p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), No. 1, 223–238.



References 105

[28] D. Cruz-Uribe, SFO, J. M. Martell and C. Perez, Sharp two-weight inequalities for

singular integrals, with applications to the Hilbert transform and the Sarason conjec-

ture. Adv. Math. 216 (2007), No. 2, 647–676.

[29] L. Diening, Maximal function on generalized Lebesgue spaces L p(·). Math. Inequal.

Appl. 7 (2004), No. 2, 245–253.

[30] L. Diening, Riesz potentials and Sobolev embeddings on generalized Lebesgue and

Sobolev spaces L p( x) and W k , p( x). Math. Nachr. 268 (2004), 31–43.

[31] L. Diening, Maximal function on Musielak-Orlicz spaces and generalized Lebesgue

spaces. Bull. Sci. Math. 129 (2005), No. 8, 657–700.

[32] L. Diening and M. Ruzicka, Calderon-Zygmund operators on generalized Lebesgue

spaces L p(·) and problems related to fluid dynamics. J. Reine Angew. Math. 63 (2003),

197–220.

[33] L. Diening and S. Samko, Hardy inequality in variable exponent Lebesgue spaces.

Frac. Calc. Appl. Anal. 10 (2007), No. 1, 1–18.

[34] M. R. Dostenic, Asymptotic behaviour of the singular value of fractional integral

operators. J. Math. Anal. Appl. 175 (1993), 380–391.

[35] O. Dragicevic, L. Grafakos, M. C. Pereyra, and S. Petermichl, Extrapolation and

sharp norm estimates for classical operators on weighted Lebesgue spaces. Publ.

Mat. 49 (2005), No. 1, 73–91.

[36] O. Dragicevic and A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via

averaging martingale transforms. Michigan Math. J. 51 (2003), No. 2, 415–435.

[37] O. Dragicevic and A. Volberg, Bellman function, Littlewood–Paley estimates and

asymptotics for the Ahlfors–Beurling operator in L p(C). Indiana Univ. Math. J. 54

(2005), No. 4, 971–995.

[38] O. Dragicevic and A. Volberg, Bellman function for the estimates of Littlewood-

Paley type and asymptotic estimates in the p − 1 problem. C. R. Math. Acad. Sci.

Paris 340 (2005), No. 10, 731–734.

[39] D. E. Edmunds and W. D. Evans, Spectral theory and differential operators. Oxford

Univ. Press, Oxford, 1987.

[40] D. E. Edmunds and W. D. Evans, Hardy Operators, Function spaces and embed-

dings. Springer, Berlin, Heidelberg, New York, 2004.

[41] D. E. Edmunds, W. D. Evans and D. J. Harris, Approximation numbers of certain

Volterra integral operators. J. London Math. Soc. 37 (1988), No. 2, 471–489.

[42] D. E. Edmunds, W. D. Evans and D. J. Harris, Two-sided estimates of the approxi-

mation numbers of certain Volterra integral operators. Studia Math. 124 (1997), No.

1, 59–80.



106 References

[43] D. E. Edmunds, A. Fiorenza and A. Meskhi, On the measure of non–compactness

for some classical operators. Acta Math. Sinica. 22 (2006), No. 6, 1847–1862.

[44] D. E. Edmunds, P. Gurka and L. Pick. Compactness of Hardy–type integral operators

in weighted Banach function spaces. Studia Math. 109 (1994), No. 1, 73–90.

[45] D.E. Edmunds, R. Kerman and J. Lang, Remainder estimates for the approximation

numbers of weighted Hardy operators acting on L2. J. Anal. Math. 85 (2001), 225–

243.

[46] D. E. Edmunds and V. Kokilashvili, Two-weight inequalities for singular integrals.

Canadian Math. Bull. 38(1995), 119–125.

[47] D. E. Edmunds and V. Kokilashvili, Two-weight compactness criteria for potential-

type operators. Proc. A. Razmadze Math. Inst. 117(1998), 123–125.

[48] D. E. Edmunds, V. Kokilashvili, A. Meskhi,Two-weight estimates for singular in-

tegrals defined on homogeneous type spaces. Canadian J. Math. 52 (2000), No. 3,

468–502.

[49] D. E. Edmunds, V. Kokilashvili, A. Meskhi, Bounded and compact integral oper-

ators. Mathematics and its Applications, 543, Kluwer Academic Publishers, Dor-

drecht, 2002.

[50] D. E. Edmunds, V. Kokilashvili and A. Meskhi, On Fourier multipliers in weighted

Triebel–Lizorkin spaces. J. Inequal. Appl. 7 (2002), No. 4, 555–591.

[51] D. E. Edmunds, V. Kokilashvili and A. Meskhi, A trace inequality for generalized po-

tentials in Lebesgue spaces with variable exponent. J. Funct. Spaces Appl., 2 (2004),

No. 1, 55–69.

[52] D. E. Edmunds, V. Kokilashvili and A. Meskhi, On the boundedness and compact-

ness of the weighted Hardy operators in L p( x) spaces. Georgian Math. J. 12 (2005),

No. 1, 27–44.

[53] D. E. Edmunds, V. Kokilashvili and A. Meskhi, Two-weight estimates in L p( x) spaces

with applications to Fourier series. Houston J. Math. (to appear).

[54] D. E. Edmunds, V. Kokilashvili and A. Meskhi, On one-sided operators in L p( x)

spaces. Math. Nachr. 281 (2008), No. 11, 1525–1548.

[55] D. E. Edmunds and J. Lang, Behaviour of the approximation numbers of a Sobolev

embedding in the one-dimensional case. J. Functional Anal. 206 (2004), 149-146.

[56] D. E. Edmunds, J. Lang and A. Nekivnda, On L p( x) norms. Proc. R. Soc. London, A

455 (1999), 219–225.

[57] D. E. Edmunds and A. Meskhi, Potential-type operators in L p( x) spaces. Z. Anal.

Anwend. 21 (2002), 681–690.



References 107

[58] D. E. Edmunds and A. Meskhi, On a measure of non–compactness for maximal

operators. Math. Nachr. 254–255 (2003), 97–106.

[59] D. E. Edmunds and J. Rakosnık, Density of smooth functions in W k , p( x)(Ω). Proc. R.

Soc. Lond. A 437 (1992), 229–236.

[60] D. E. Edmunds and J. Rakosnık, Sobolev embeddings with variable exponent. Studia

Math. 143 (2000), 267–293.

[61] D. E. Edmunds and J. Rakosnık, Sobolev embeddings with variable exponent II.

Math. Nachr., 246-247 (2002), 53–67.

[62] D. E. Edmunds and V. D. Stepanov, The measure of noncompactness and approxima-

tion numbers of certain Volterra integral operators. Math. Ann. 298 (1994), 41–66.

[63] D. E. Edmunds and V. D. Stepanov, On the singular numbers of certain Volterra

integral operators. J. Funct. Anal. 134 (1995), No. 1, 222–246.

[64] D. E. Edmunds and H. Triebel, Function spaces, entropy numbers, differential oper-

ators. Cambridge Univ. Press, Cambridge, 1996.

[65] W. D. Evans, D. J. Harris, J. Lang, Two-sided estimates for the approximation num-

bers of Hardy-type operators in L∞ and L1. Studia Math. 130 (1998), No. 2, 171-192.

[66] W. D. Evans, D. J. Harris and J. Lang, The approximation numbers of Hardy-type

operators on trees. Proc. London Math. Soc. 83 (2001), No. 2, 390–418.

[67] V. Faber and G. M. Wing, Singular values of fractional integral operators: a unifi-

cation of theorems of Hille, Tamarkin, and Chang. J. Math. Anal. Appl. 120 (1986),

745–760.

[68] I. Feldman, N. Krupnik and I. Spitkovsky, Norms of the singular integral operators

with Cauchy kernel along certain. Integral Equations Operator Theory 24 (1996),

68–80.

[69] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley theory and the study of func-

tion spaces. Regional Conference Series in Mathematics, Vol. 79, Amer. Math. Soc.Providence, RI, 1991.

[70] G. B. Folland, E. M. Stein, Hardy spaces on homogeneous groups. Princeton Univ.

Press, Princeton, New Jersey, 1982

[71] M. Gabidzashvili, Weighted inequalities for anizotropic potentials (Russian). Trudy

Tbiliss. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 89 (1986), 25–36.

[72] M. Gabidzashvili and V. Kokilashvili, Two-weight weak type inequalities for frac-

tional type integrals. Preprint , No. 45, Mathematical Institute Czech Acad. Sci.,

Prague 1989.



108 References

[73] J. Garcıa–Cuerva J. and J. L. Rubio de Francia, Weighted norm inequalities and

related topics. North-Holland Mathematics Studies Vol. 116, Mathematical Notes

Series 104, North-Holland Publishing Co., Amsterdam, 1985.

[74] I. Genebashvili, Generalized fractional maximal functions and potentials in weighted

spaces. Proc. A. Razmadze Math. Inst. 102 (1993), 41–57.

[75] I. Genebashvili, A. Gogatishvili and V. Kokilashvili, Solution of two-weight prob-

lems for integral transforms with positive kernels. Georgian Math. J. 3 (1996), No.

1, 319–342.

[76] I. Genebashvili, A. Gogatishvili, V. Kokilashvili and M. Krbec, Weight theory for in-

tegral transforms on spaces of homogeneous type. Pitman Monographs and Surveys

in Pure and Applied Mathematics, 92, Longman, Harlow, 1998.

[77] D. Gluskin, Norms of random matrices and diameters of finite-dimensional sets.

Math. USSR Sb., 184(1984), 173–182.

[78] A. Gogatishvili and V. Kokilashvili, Criteria of two–weight strong type inequalities

for fractional maximal functions. Georgian Math. J. 3 (1996), No. 5, 423–446.

[79] I. Gohberg and N. Krupnik, On the spectrum of singular integral operators on the

spaces L p (Russian). Studia Math. 31(1968), 347–362.

[80] I. Gohberg and N. Krupnik, On the norm of the Hilbert transform on the space L p

(Russian). Funktsional Anal. i. Prilozh. 2 (1968), No. 2, 91–92.

[81] I. Gohberg and N. Krupnik, One-dimensional linear singular integrals equations.

Vol. 1,2, Birkhauser Verlag, Basel, Boston, Berlin, 1992.

[82] R. Gorenflo and S. Samko, On the dependence of asymptotics of s-numbers of frac-

tional integration operators on weight functions. Integral Transforms Spec. Funct . 5

(1997), No. 3–4, 191–212.

[83] R. Gorenflo and Vu Kim Tuan, Asymptotics of singular values of fractional integra-

tion operators. Inverse Problems 10 (1994), No. 4, 949–955.

[84] L. Grafakos and S. Montgomery-Smith, Best constants for uncentred maximal func-tions. Bull. London Math. Soc. 29 (1997), No. 1, 60–64

[85] L. Grafakos, S. Montgomery-Smith and O. Motrunich, A sharp estimate for the

Hardy-Littlewood maximal function. Studia Math. 134 (1999), No. 1, 57–67.

[86] V. Guliev, Two–weight L p inequality for singular integral operator on Heisenberg

groups. Georgian Math. J. 1(1994), No. 4, 367–376.

[87] V. Guliev, Integral operators, function spaces and questions of approximation on

Heisenberg groups. Baku- ”ELM”, 1996.

[88] G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities. Cambridge Univ. Press,

1934.



References 109

[89] P. Harjulehto, P. Hasto and M. Pere, Variable exponent Lebesgue spaces on metric

spaces: the Hardy-Littlewood maximal operator. Real Anal. Exchange. 30 (2004-

2005), 87–104.

[90] P. Harjulehto, P. Hasto, M. Koskenoja, Hardy’s inequality in a variable exponent

Sobolev space. Georgian Math. J. 12 (2005), No. 3, 431–442.

[91] P. Hasto, On the density of continuous functions in variable exponent Sobolev space.

Rev. Mat. Iberoam. 23 (2007), No. 1, 213–234.

[92] S. Hencl, Measure of non–compactness of classical embeddings of Sobolev spaces.

Math. Nachr , 258(2003), 28–43.

[93] L. Hormander, L p estimates for (pluri-) subharmonic functions. Math. Scand.

20(1967), 65–78.

[94] H. Hudzik, On generalized Orlicz-Sobolev space. Funct. Approx. Comment. Math. 4(1976), 37–51.

[95] R. A. Hunt, B. Muckenoupt and R.L. Wheeden, Weighted norm inequalities for the

conjugate function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227–

251.

[96] T. Iwaniec, Extremal inequalities in Sobolev spaces and quasiconformal mappings.

Z. Anal. Anwendungen 1(1982), 1–16.

[97] P. Jain, P. K. Jain and B. Gupta, Higher dimensional compactness of Hardy operatorsinvolving Oinarov-type kernels. Math. Ineq. Appl.,9 (2002), No. 4, 739–748.

[98] R. K. Juberg, The measure of noncompactness in L p for a class of integral operators.

Indiana Univ. Math. J. 23 (1974), 925–936.

[99] R. K. Juberg, Measure of noncompactness and interpolation of compactness for a

class of integral transformations. Duke Math. J. 41 (1974), 511–525.

[100] V. P. Kabaila, On imbedding the space L p( µ) into Lr ( ν). Litovsk. Mat. Sb. 21 (1981),

No. 4, 143–148.

[101] L. P. Kantorovich and G. P. Akilov, Functional Analysis. Pergamon, Oxford, 1982.

[102] A. Yu. Karlovich, On the essential norm of the Cauchy singular integral operator

in weighted rearrangement-invariant spaces. Integral Equations Operator Theory 38

(2000), No. 1, 28–50.

[103] A. Yu. Karlovich, Fredholmness of singular integral operators with piecewise con-

tinuous coefficients on weighted Banach function spaces. J. Integral Equations Appl.

15 (2003), No. 3, 263–320

[104] B. S. Kashin, Diameters of some finite dimensional sets and some classes of smooth

functions (Russian). Izv. Akad. Nauk SSSR, Ser. Mat. 41 (1977), 334–351.



110 References

[105] M. Khabazi, Maximal operators in weighted L p(·) spaces. Proc. A. Razmadze Math.

Inst. 135 (2004), 143–144.

[106] M. Khabazi, Maximal functions in L p(·) spaces. Proc. A. Razmadze Math. Inst. 135

(2004), 145–146.

[107] G. Khuskivadze, V. Kokilashvili and V. Paatashvili, Boundary value problems for

analytic and harmonic functions in domains with nonsmooth boundaries. Applica-

tions to conformal mappings. Memoirs on Differential Equations and Math. Phys.

14 (1998), 1–195.

[108] V. M. Kokilashvili, On Hardy’s inequalities in weighted spaces (Russian). Soobsch.

Akad. Nauk Gruz. SSR 96 (1979), 37–40.

[109] V. Kokilashvili, On singular and bisingular integral operators in weighted spaces

(Russian). Proc. Mat. Inst. Razmadze Akad. Nauk Gruzin. SSR 19 (1982), 51–79.

[110] V. Kokilashvili, New aspects in weight theory. In: Function Spaces, Differential

Operators and Nonlinear Analysis. Prometeus Publishing House, Prague, 1996, 51–

70.

[111] V. Kokilashvili, On a progress in the theory of integral operators in weighted Banach

Function Spaces. In ”Function Spaces, Differential Operators and Nonlinear Anal-

ysis”, Proceedings of the Conference held in Milovy, Bohemian-Moravian Uplands,

May 28– June 2, Math. Inst. Acad. Sci. of Czech Republic, Prague, 2004.

[112] V. Kokilashvili and M. Krbec, Weighted inequalities in Lorentz and Orlicz spaces.Singapore, New Jersey, London, Hong Kong: World Scientific, 1991.

[113] V. Kokilashvili and A. Meskhi, Two–weight estimates for singular integrals defined

on homogeneous groups. (Russian) Doklady Acad. Nauk 354 (1997), No 3. 301–303.

English transl. in Doklady Mathematics, 55 (1997), No. 3, 362–364.

[114] V. Kokilasvili and A. Meskhi, Two weight inequalities for singular integrals defined

on homogeneous groups. Proc. A. Razmadze Math. Inst. Georgian Acad. Sci. 112

(1997), 57–90.

[115] V. Kokilashvili and A. Meskhi, Weighted inequalities for the Hilbert transform andmultipliers of Fourier transforms. J. Ineq. Appl. 1 (1997), No. 3, 239–252.

[116] V. Kokilashvili and A. Meskhi, Boundedness and compactness criteria for some clas-

sical integral operators. In: Lecture Notes in Pure and Applied Mathematics, 213, ”

Function Spaces V”, Proceedings of the Conference, Pozna´ n, Poland ( Ed. H. Hudzik

and L. Skrzypczak ), 279–296, New York, Bazel, Marcel Dekker, 2000.

[117] V. Kokilasvili and A. Meskhi, Fractional integrals on measure spaces. Frac. Calc.

Appl. Anal. 4 (2001), No. 1, 1–24.

[118] V. Kokilashvili and A. Meskhi, On a trace inequality for one–sided potentials and

applications to the solvability of nonlinear integral equations. Georgian Math. J. 8

(2001), No. 3, 521–536.



References 111

[119] V. Kokilashvili and A. Meskhi, Weighted criteria for generalized fractional maximal

functions and potentials in Lebesgue spaces with variable exponent. Integral Trans-

forms Spec. Funct. 18 (2007), No. 9, 609–628.

[120] V. Kokilashvili, V. Paatashvili and S. Samko, Boundary value problems for analytic

functions in the class of Cauchy-type integrals with density in L p(·)

. Bound. ValueProbl. 2005, No. 1, 43-71.

[121] V. Kokilashvili, V. Paatashvili and S. Samko, Boundedness in Lebesgue spaces with

variable exponent of the Cauchy singular operator on Carleson curves. In: Ya. Erusal-

imsky , I. Gohberg, S. Grudsky, V. Rabinovich, and N. Vasilevski, Eds, “Operator

Theory: Advances and Applications”, Dedicated to 70-th Birthday of Prof. I. B. Si-

monenko, Vol. 170, 167-186, Birkhauser Verlag, Basel, 2006.

[122] V. Kokilashvili and S. Samko, Maximal and fractional operators in weighted L p( x)

spaces. Rev. Mat. Iberoamericana 20 (2004), No. 2, 493–515.

[123] V. Kokilashvili and S. Samko, On Sobolev theorem for Riesz-type potentials in

Lebesgue spaces with variable exponent. Z. Anal. Anwendungen 22 (2003), No. 4,

899–910.

[124] V. Kokilashvili and S. Samko, Singular integrals in weighted Lebesgue spaces with

variable exponent. Georgian Math. J. 10 (2003), No. 1, 145–156.

[125] V. Kokilashvili and S. Samko, Sobolev theorem for potentials on Carleson curves

in variable Lebesgue spaces. Mem. Differential Equations Math. Phys. 33 (2004),157–158.

[126] V. Kokilashvili and S. Samko, Boundedness of maximal operators and potential op-

ertators on Carleson curves in Lebesgue spaces with variable exponent. Acta Math.

Sin. ( Engl. Ser.) 24 (2008), No. 11, 1775–1800.

[127] V. Kokilashvili and S. Samko, Vekua’s generalized singular integrals on Carleson

curves in weighted Lebesgue spaces. Operator Theory: Advances and Applications

181 (2008).

[128] V. Kokilashvili and S. Samko, Weighted boundedness in Lebesgue spaces with vari-

able exponents of classical operators on Carleson curves. Proc. A. Razmadze Math.

Inst. 138 (2005), 106–110.

[129] V. Kokilashvili and S. Samko, Boundedness in Lebesgue spaces with variable

exponent of maximal, singular and potential operators. Izvestija VUZov. Severo-

Kavkazskii Region. Estestvennie Nauki. 2005, Special issue “Pseudodifferential

equations and some problems of mathematical physics”, dedicated to 70th birthday

of Prof. I.B.Simonenko, 152–158.

[130] V. Kokilashvili, S. Samko, Operators of harmonic analysis in weighted spaces with

non-standard growth. J. Math. Anal. Appl. (2008) doi: 10.1016/j.jmaa. 2008.06.056.



112 References

[131] V. Kokilashvili, N. Samko, and S. Samko, The maximal operator in variable spaces

L p(·)(Ω,ρ). Georgian Math. J. 13 (2006), No.1, 109–125.

[132] V. Kokilashvili, N. Samko, and S. Samko, Singular operators in variable spaces

L p(·)(Ω,ρ) with oscillating weights. Math. Nachr. 280 (9-10) (2007), 1145–1156.

[133] V. Kokilashvili, N. Samko, and S. Samko, The maximal operator in weighted variable

spaces L p(·). J. Function spaces and Appl. 5 (2007), No. 3, 299–317.

[134] V. Kokilashvili, N. Samko and S. Samko, Boundedness of the generalized singular

integral operator on Carleson curves in weighted variable Lebesgue spaces. Proc. A.

Razmadze Math. Inst. 142 (2006), 138–142.

[135] H. Konig, Eigenvalue distribution of compact operators. Birkhauser, Boston, 1986.

[136] T. S. Kopaliani, On some structural properties of Banach function spaces and bound-

edness of certain integral operators. Czechoslovak Math. J. 54 (2004), No. 3, 791–805.

[137] T. S. Kopaliani, Infirmal convolution and Muckenhoupt A p(·) condition in variable

L p spaces. Arch. Math. 89 (2007), 185–192.

[138] O. Kovacik and J. Rakosnık, On spaces L p( x) and W k , p( x), Czechoslovak Math. J. 41

(116) (1991), No. 4, 592–618.

[139] M. A. Krasnoselskii, P. P. Zabreiko, E. I. Pustilnik and P. E. Sobolevskii, Integral

operators in spaces of summable functions. (Russian) (Nauka, Moscow, 1966), Engl.

Transl. Noordhoff International Publishing, Leiden, 1976.

[140] M. Krbec, B. Opic, L. Pick and J. Rakosnık, Some recent results on Hardy type oper-

ators in weighted function spaces and related topics. In: Function spaces, Differential

Operators and Nonlinear Analysis (Friedrichroda, 1992), 158–184, Teubner-Texte

Math., 133, Teubner, Stuttgart , 1993.

[141] N. Krupnik and I. Verbitsky, Exact constants in the theorems of K. I. Babenko and B.

V. Khvedelidze on the boundedness of singular operators. (Russian) Soobshch. AN

Gruz. SSR 85 (1977), No. 1, 21–24.

[142] N. Krupnik and I. Verbitsky, Exact constants in theorems on the boundedness of

singular operators in L p spaces with a weight and their application. (Russian) Linear

operators. Mat. Issled . 54 (1980), 21–35, 165.

[143] A. Kufner, O. John and S. Fucık, Function spaces. Monographs and Textbooks on

Mechanics of Solids and Fluids; Mechanics: Analysis. Noordhoff International Pub-

lishing, Leyden; Academia, Prague, 1977.

[144] A. Kufner, L. Maligranda, L.-E. Persson, The Hardy inequality. About its history and

some related results. Vydavatelsk y Series, Plzen, 2007.

[145] A. Kufner and L.-E. Persson, Weighted inequalities of Hardy type. World Scientific

Publishing Co., Inc., River Edge, NJ, 2003.



References 113

[146] T. Kuhn, Entropy numbers of diagonal operators of logarithmic type. Georgian Math.

J. 8 (2001), No. 2, 307–318.

[147] M. T. Lacey, E. T. Sawyer and I. Uriarte-Tuero, A charachterization of two weight

inequalities for maximal singular integrals. Preprint , arXiv:0807.0246v1 [math.CA],

1 July, 2008.

[148] J. Lang, Improved estimates for the approximation numbers of Hardy-type operators.

J. Approx. Theory, 121 (2003), 61–70.

[149] M. A. Lifshits and W. Linde, Approximation and entropy numbers of Volterra oper-

ators with application to Brownian motion. Mem. Amer. Math. Soc., Vol. 157, No.

745, 2002.

[150] M. Lindstrom and E. Wolf, Essential norm of the difference of weighted composition

operators. Monatsh. Math. 153

(2008), 133–143.

[151] P. I. Lizorkin and M. O. Otelbaev, Estimates of approximation numbers of embed-

dings of the Sobolev spaces with weights. Proc. Steklov Inst. Math. 170 (1984), 213–

232.

[152] E. Lomakina and V. Stepanov, On asymptotic behavior of the approximation numbers

and estimates of Schatten- von Neumann norms of hardy-type integral operators. In:

Function Spaces and Applications. New Delhi: Narosa Publishing House, 2000,

153–187.

[153] M. Lorente, A characterization of two–weight norm inequalities for one–sided oper-

ators of fractional type. Canad. J. Math. 49 (1997), No. 5, 1010–1033.

[154] M. Lorente and A. de la Torre, Weighted inequalities for some one–sided operators.

Proc. Amer. Math. Soc. 124 (1996), 839–848.

[155] F. J. Martin–Reyes, Weights, one–sided operators, singular integrals, and ergodic

theorems. In: Nonlinear Analysis, Function Spaces and Applications, Vol 5, Pro-

ceedings, M. Krbec et. al. (eds), Prometheus Publishing House, Prague, 1994, 103–

138.

[156] F. J. Martin–Reyes and E. Sawyer, Weighted inequalities for Riemann–Liouville

fractional integrals of order one and greater. Proc. Amer. Math. Soc. 106 (1989),

727–733.

[157] F. J. Martin–Reyes and A. de la Torre, Two–weight norm inequalities for fractional

one–sided maximal operators. Proc. Amer. Math. Soc. 117 (1993), 483–489.

[158] V. G. Maz’ya , Sobolev spaces. Springer–Verlag, Berlin–Heidelberg, 1985.

[159] V. G. Maz’ya, I. E. Verbitsky, Capacitary inequalities for fractional integrals, withapplications to partial differential equations and Sobolev multipliers. Ark. Mat. 33

(1995), 81–115.



114 References

[160] A. Meskhi, Solution of some weight problems for the Riemann-Liouville and Weyl

operators. Georgian Math. J. 5 (1998), No. 6, 565–574.

[161] A. Meskhi, On the measure of non–compactness and singular numbers for the

Volterra integral operators. Proc. A. Razmadze Math. Inst. 123 (2000), 162–165.

[162] A. Meskhi, On the singular numbers for some integral operators. Revista Mat. Comp.

14 (2001), No. 2, 379–393.

[163] A. Meskhi, Criteria for the boundedness and compactness of integral transforms with

positive kernels. Proc. Edinb. Math. Soc., 44 (2001), No. 2, 267–284.

[164] A. Meskhi, Asymptotic behaviour of singular and entropy numbers for some

Riemann–Liouville type operators. Georgian Math. J. 8 (2001), No. 2, 157–159.

[165] A. Meskhi, On a measure of non–compactness for the Riesz transforms. Proc. A.

Razmadze Math. Inst. 129 (2002), 156–157.

[166] A. Meskhi, On a measure of non–compactness for singular integrals. J. Function

Spaces Appl. 1 (2003), No. 1, 35–43.

[167] Y. Mizuta and T. Shimomura, Sobolev’s inequality for Riesz potentials with vari-

able exponent satisfying a log–Holder condition at infinity. J. Math. Anal. Appl. 311

(2005), 268–288.

[168] B. Muckenhoupt, Hardy’s inequality with weights. Studia Math. 44 (1972), 31–38.

[169] B. Muckenhoupt, Weight norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226.

[170] B. Muckenhoupt, Two–weight norm inequalities for the Poisson integral.

Trans. Amer. Math. Soc. 210 (1975), 225–231.

[171] B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for fractional inte-

grals. Trans. Amer. Math. Soc. 192 (1974), 261–276.

[172] B. Muckenhout and R. L. Wheeden, Two–weight function norm inequalities for

the Hardy–Littlewood maximal function and the Hilbert transform. Studia Math. 55(1976), No. 3, 279–294.

[173] J. Musielak, Orlicz spaces and modular spaces. Lecture Notes in Math., Vol. 1034,

Springer-Verlag, Berlin, 1983.

[174] J. Musielak and W. Orlicz, On modular spaces. Studia Math. 18 (1959), 49–65.

[175] K. T. Mynbaev and M. O. Otelbaev, Weighted function spaces and the spectrum of

differential operators (Russian). Nauka, Moscow, 1988.

[176] M. Nakano, Topology of linear topological spaces. Moruzen Co. Ltd, Tokyo, 1981.

[177] F. Nazarov and A.Volberg, Heat extension of the Beurling operator and estimates for

its norm. St. Petersburg Math. J. 15 (2004), No. 4, 563–573.



References 115

[178] F. Nazarov, S.Treil and A. Volberg, Two-weight estimate for the Hilbert transform

and corona decomposition for non doubling measures. Preprint, 2004, pp. 1–38.

[179] F. Nazarov, S.Treil and A. Volberg, Two weight T 1 theorems for the Hilbert trans-

form: the case of doubling measures. Preprint, 2004, pp. 1–40.

[180] A. Nekvinda, Hardy-Littlewood maximal operator on L p(·)(Rn). Math. Ineq. Appl. 7

(2004), No. 2, 255–265.

[181] J. Newman and M. Solomyak, Two–sided estimates on singular values for a class of

integral operators on the semi–axis. Integral Equations Operator Theory 20 (1994),

335–349.

[182] K. Nowak, Schatten ideal behavior of a generalized Hardy operator. Proc. Amer.

Math. Soc. 118 (1993), 479–483.

[183] R. Oinarov, Two-sided estimate of certain classes of integral operators (Russian). Tr.

Math. Inst. Steklov. 204 (1993), 240–250.

[184] G. G. Oniani, On the measure of non–compactness of maximal operators. J. Funct.

Spaces Appl. 2 (2004), No. 2, 217–225.

[185] G. G. Oniani, On the non-compactness of maximal operators. Real Anal. Exchange

28 (2002/03), No. 2, 439–446.

[186] G. G. Oniani, On non-compact operators in weighted ideal and symmetric function

spaces. Georgian Math. J. 13 (2006) No. 3, 501–514.

[187] B. Opic, On the distance of the Riemann–Liouville operators from compact opera-

tors. Proc. Amer. Math. Soc. 122 (1994), No. 2, 495–501.

[188] B. Opic and A. Kufner, Hardy-type inequalities. Pitman Research Notes in Mathe-

matics. Series 219, Longman Scientific & Technical, Harlow, 1990.

[189] W. Orlicz, Uber konjugierte exponentenfolgen. Studia Math. 3 (1931), 200–211.

[190] C. Perez, Weighted norm inequalities for singular integral operators. J. London Math.

Soc. 2(49) (1994), 296–308.

[191] C. Perez, Banach function spaces and the two–weight problem for maximal func-

tions. In: Function Spaces, Differential Operators and Nonlinear Analysis, M. Krbec

et. al. (eds), Paseky nad Jizerou, 1995, 1–16, Prometheus Publishing House, Prague,

1996, 141–158.

[192] S. Petermichl, The sharp weighted bound for the Riesz transforms. Proc. Amer. Math.

Soc. 136 (2008), No. 4, 1237–1249.

[193] S. Petermichl, The sharp bound for the Hilbert transform on weighted Lebesguespaces in terms of the classical A p characteristic. Amer. J. Math. 129 (2007), No. 5,

1355–1375.



116 References

[194] S. Petermichl and J. Wittwer, A sharp weighted estimate on the norm of Hilbert

transform via invariant A2 characteristic of the weight. Michgan Math. J. 50 (2002),

71–87.

[195] S. K. Pichorides, On the best values of the constants in the theorems of M. Riesz,

Zygmund and Kolmogorov. Studia Math. 44 (1972), 165–179.

[196] L. Pick and M. Ruzicka, An example of a space L p( x) on which the Hardy-Littlewood

maximal operator is not bounded. Expo. Math. 19 (2001), 369–371.

[197] A. Pietsch, Eigenvalues and s–numbers. Cambridge Univ. Press, Cambridge, 1987.

[198] D. V. Prokhorov, On the boundedness of a class of integral operators. J. London

Math. Soc., 61 (2000), No. 2, 617–628.

[199] D. V. Prokhorov and V. Stepanov, Weighted estimates for the Riemann– Liouville

operators and applications. Trudy Mat.Inst. Steklov. 243 (2003), 289–312.

[200] H. Rafeiro and S. Samko, Dominated compactness theorem in Banach function

spaces and its applications. Compl. Anal. Oper. Theory 2 (2008), No. 4, 669–681.

[201] Y. Rakotontratsimba, Two-weight norm inequality for Calderon-Zygmund operators.

Acta Math. Hungar. 80 (1998), 39–54.

[202] M. Ruzicka, Electrorheological fluids: modeling and mathematical theory. Lecture

Notes in Mathematics, Vol. 1748, Springer-Verlag, Berlin, 2000.

[203] S. G. Samko, Differentiation and integration of variable order and the spaces L p( x).

Proceed. of Intern. Conference “Operator Theory and Complex and Hypercomplex

Analysis”, 12–17 December 1994, Mexico City, Mexico, Contemp. Math., 1997.

[204] S. Samko, Convolution type operators in L p( x). Integral Transforms Spec. Funct. 7

(1998), No. 1–2, 123–144.

[205] S. Samko, Convolution and potential type operators in L p( x)(Rn). Integral Transforms

Spec. Funct. 7 (1998), No. 3–4, 261–284.

[206] S. Samko, Denseness of C ∞0 ( R

n

) in the generalized Sobolev spaces W m, p( x)

( Rn

). Di-rect and Inverse Problems of Mathematical Physics (Newark, DE, 1997), 333–342,

Int. Soc. Anal. Appl. Comput ., 5, Kluwer Acad. Publ., Dordrecht, 2000.

[207] S. G. Samko, Hypersingular integrals and their applications. Analytical Methods and

Special Functions, 5, Taylor & Francis, Ltd., London, 2002.

[208] S. Samko, On a progress in the theory of Lebesgue spaces with variable exponent:

maximal and singular operators. Integral Transforms Spec. Funct. 16 (2005), No.

5-6, 461–482.

[209] S. G. Samko, A. A. Kilbas, O. I. Marichev, Fractional Integrals and Derivatives.

Theory and Applications. Gordon and Breach Sci. Publishers, London–New York,

1993.



References 117

[210] S. Samko, E. Shargorodsky and B. Vakulov, Weighted Sobolev theorem with variable

exponent for spatial and spherical potential operators II. J. Math. Anal. Appl. 325

(2007), No. 1, 745–751.

[211] S. Samko and B. Vakulov, Weighted Sobolev theorem with variable exponent. J.

Math. Anal. Appl. 310 (2005), 229–246.

[212] E. T. Sawyer, A characterization of a two-weight norm inequality for maximal oper-

ators. Studia Math. 75 (1982), 1–11.

[213] E. T. Sawyer, A two-weight weak type inequality for fractional integrals. Trans.

Amer. Math. Soc. 281 (1884), 399–345.

[214] E. T. Sawyer, Two-weight norm inequalities for certain maximal and integral oper-

ators. In: Harmonic analysis, Minneapolis, Minn. 1981, 102–127, Lecture Notes in

Math. 908, Springer, Berlin, New York,

1982.

[215] E. T. Sawyer, Multipliers of Besov and power weighted L2 spaces. Indiana Univ.

Math. J. 33(3)(1984), 353–366.

[216] E. Sawyer, Weighted inequalities for the one-sided Hardy-Littlewood maximal func-

tions. Trans. Amer. Math. Soc. 297 (1986), 53–61.

[217] E. T. Sawyer and R. L. Wheeden, Weighted inequalities for fractional integrals on

Euclidean and homogeneous spaces. Amer. J. Math. 114 (1992), 813–874.

[218] E. T. Sawyer and R. L. Wheeden, Carleson conditions for the Poisson integrals. In-diana Univ. Math. J. 40 (1991), No. 2, 639–676.

[219] E. T. Sawyer, R. L. Wheeden and S. Zhao, Weighted norm inequalities for operators

of potential type and fractional maximal functions. Potential Anal., 5 (1996), 523–

580.

[220] I. I. Sharapudinov, On a topology of the space L p( x)([0,1]). Mat. Zam. 26 (1979), No.

4, 613–632.

[221] G. Sinnamon, One-dimensional Hardy-inequalities in many dimensions. Proc. RoyalSoc. Edinburgh Sect, A, 128A (1998), 833–848.

[222] G. Sinnamon and V. Stepanov, The weighted Hardy inequality: new proof and the

case p = 1. J. London Math. Soc., 54 (1996), 89–101.

[223] M. Solomyak, Estimates for the approximation numbers of the weighted Riemann–

Liouville operator in the space L p. Oper. Theory, Adv. Appl. 113 (2000), 371–383.

[224] E. M. Stein, Harmonic analysis: real variable methods, orthogonality and oscilla-

tory integrals. Princeton University Press, Princeton, New Jersey, 1993.

[225] V. Stepanov. Two-weight estimates for the Riemann-Liouville operators (Russian).

Izv. Akad. Nauk SSSR., 54 (1990), No. 3, 645–656.



118 References

[226] V. D. Stepanov, Weighted norm inequalities for integral operators and related top-

ics. In: Nonlinear Analysis, Function spaces and Applications, 5, 139–176, Olimpia

Press, Prague, 1994.

[227] J. O. Stromberg and A. Torchinsky, Weighted Hardy spaces. Lecture Notes in Math.,

Vol. 1381, Springer Verlag, Berlin, 1989.

[228] C. A. Stuart, The measure of non–compactness of some linear integral operators.

Proc. R. Soc. Edinburgh A71 (1973), 167–179.

[229] H. Triebel, Interpolation theory, function spaces, differential operators. North–

Holland, Amsterdam, 1978 (Second ed. Barth, Heidelberg, 1995).

[230] B. O. Turesson, Non-linear potential theory and weighted Sobolev spaces. Lecture

Notes in Math., Vol. 1736, Springer, Berlin, 2000.

[231] I. E. Verbitsky, Weighted norm inequalities for maximal operators and Pisier’s the-orem on Factorization through L p∞. Integral equations Operator Theory. 15 (1992),

No. 1, 124–153.

[232] I. E. Verbitsky and R.L. Wheeden, Weighted estimates for fractional integrals and

applications to semilinear equatuions. J. Funct. Anal. 129 (1995), 221–241.

[233] A. Volberg, Calder on–Zygmund capacities and operators on nonhomogeneous

spaces. CBMS Regional Conference Series in Mathematics, Vol. 100, 2003.

[234] A. Wedestig, Weighted inequalities of Hardy-type and their limiting inequalities.Doctoral Thesis, Lulea University of Technology, Department of Mathematics, 2003.

[235] G. Welland. Weighted norm inequalities for fractional integrals. Proc. Amer. Math.

Soc. 51 (1975), 143–148.

[236] R. L. Wheeden, A characterization of some weighted norm inequalities for the frac-

tional maximal functions. Studia Math. 107 (1993), 251–272.

[237] N. A. Yerzakova, The measure of noncompactness of Sobolev embeddings. Integral

Equations Operator Theory 18 (1994), No. 3, 349–359.

[238] V. V. Zhikov, Averaging of functionals of the calculus of variations and elasticity

theory. Math. USSR Izv. 29 (1987), 33–66.

[239] V. V. Zhikov, On Lavrentiev’s phenomenon. Russian J. Math. Phys. 3 (1995), 249–

269.

[240] V. V. Zhikov, On some variational problems. Russian J. Math. Phys. 5 (1997), 105–

116.

[241] W. P. Ziemer, Weakly differentiable functions. Springer-Verlag, New York, 1983.

[242] A. Zygmund, Trionometric Series. Vol. I, Cambridge Univ. Press, Cambridge, 1959.



Index

A

Adams, 103

AMS, 104Amsterdam, 108, 118

application, 112, 113applied mathematics, ix

asymptotic, 25, 105, 113asymptotics, 105, 108

averaging, 105

B

Banach spaces, vii, 4, 16, 63, 80

behavior, 113, 115Boston, 103, 108, 112

boundary value problem, vii, viii bounded linear operators, 4, 22, 67

Brownian motion, 113

C

calculus, vii, ix, 118Cauchy integral, 26, 83

classes, vii, 4, 5, 109, 115

classical, 16, 25, 26, 74, 82, 83, 102, 104, 105, 106,109, 110, 111, 115

closure, 63

composition, 113conjecture, 105

continuity, 7, 13, 15, 78corona, 115

Czech Republic, 110

D

decomposition, 115

definition, 1, 19, 29, 36, 37, 83, 86, 90density, 109, 111

derivatives, xi, 63differential equations, viiidifferentiation, viii, 28, 33, 35

distribution, 112

duality, 54, 84

E

economics, viiielasticity, 118

encouragement, ixentropy, 4, 25, 67, 107, 113, 114

equality, 75, 91, 94, 99Euclidean space, viii, xi, 9, 35, 51

F

family, 6finance, viii

fluid, 105Fourier, 51, 68, 70, 106, 110

fractional integrals, vii, 50, 113, 114, 117, 118

G

gene, 104generalization, 17

generalizations, 104graduate students, ix

groups, viii, 1, 6, 35, 37, 51, 102, 103, 107, 108, 110

growth, viii, 83, 111

H

Harmonic analysis, 117Heisenberg, viii, 1, 108

Heisenberg group, viii, 1, 108

Hilbert, vii, viii, 14, 22, 23, 24, 26, 36, 83, 88, 91,102, 103, 105, 108, 109, 110, 114, 115, 116

Hilbert space, 22

Holland, 118Hong Kong, 110

House, 110, 113, 115

I

identity, vii, viii, 25, 51, 63, 65



Index120

Indiana, 105, 109, 117

inequality, 2, 3, 5, 6, 8, 9, 10, 11, 12, 13, 14, 15, 16,17, 18, 19, 23, 24, 25, 31, 34, 35, 37, 39, 40, 44,

47, 49, 51, 52, 54, 55, 56, 58, 59, 60, 61, 62, 63,

64, 66, 67, 69, 71, 73, 74, 75, 76, 78, 79, 80, 84,85, 86, 87, 88, 90, 92, 93, 94, 95, 96, 98, 99, 100,

103, 104, 105, 106, 108, 109, 110, 112, 114, 116,

117integration, viii, 42, 108, 116

interval, 12, 13, 20, 21, 71, 80, 93

J

Jordan, 23, 86, 87, 98

K

kernel, vii, 21, 24, 25, 37, 38, 70, 103, 107Kolmogorov, 116

L

lead, ix, 96Lebesgue measure, xi

Lie algebra, 1

Lie group, 1linear, 1, 3, 4, 5, 7, 8, 9, 10, 14, 16, 17, 22, 60, 67,

80, 81, 93, 108, 114, 118

linear function, 3, 14, 16, 17, 60London, 104, 105, 106, 107, 108, 110, 115, 116, 117

Lyapunov, 102

M

manifold, viii

manifolds, viiimartingale, 36, 105

mathematicians, ixmathematics, vii

measures, 115

memory, viiimemory processes, viii

metric, 70, 109

Mexico, 116Mexico City, 116

modeling, 116monograph, vii, viii, ix, 1, 25, 70

Moscow, 112, 114motion, 113

N

natural, vii, ix, xi

New Jersey, 107, 110, 117

New York, 105, 110, 117, 118

nonlinear, vii, 110non-linearity, ix

norms, 25, 37, 45, 52, 104, 106, 113

numerical analysis, viii

O

operator, vii, 1, 3, 4, 5, 7, 8, 9, 10, 11, 12, 13, 16, 17,21, 22, 23, 24, 25, 26, 27, 28, 30, 31, 33, 36, 37,

43, 45, 47, 50, 51, 54, 55, 61, 62, 63, 64, 65, 71,

77, 78, 79, 80, 82, 86, 87, 88, 90, 98, 99, 100, 102,104, 105, 108, 109, 111, 112, 114, 115, 116, 117

Operators, 27, 29, 31, 33, 35, 37, 39, 41, 43, 45, 47,49, 51, 53, 55, 57, 59, 61, 63, 65, 67, 69, 90, 105,

110, 111, 112, 115organization, ix

orthogonality, 117

P

paper, ix, 13, 25, 69, 70

parameter, 23, 98Paris, 105

partial differential equations, 113 physics, vii, viii, 111

plasticity, ixPoisson, 51, 60, 70, 114, 117

Poland, 110

power, 117 property, 4

Q

quantum, vii

quantum mechanics, vii

R

radius, xi, 1, 5, 6, 14, 57, 60, 67

random, 108random matrices, 108

range, vii

real numbers, 9, 17recall, 35, 74, 94

research, ixresearchers, viii, ix

Russian, 104, 107, 108, 109, 110, 112, 114, 115,117, 118

S

series, 18, 51, 68, 70, 106

Singapore, 110



Index 121

singular, vii, viii, 1, 22, 23, 25, 26, 83, 86, 90, 98,

102, 104, 105, 106, 107, 108, 109, 110, 111, 112,113, 114, 115, 116

Sobolev space, viii, ix, 25, 63, 105, 109, 113, 116,

118solutions, vii

spatial, 117

spectrum, 104, 108, 114St. Petersburg, 114

stochastic, vii, viiistochastic processes, vii

students, ixsymbols, 104

systems, 22, 46

T

theory, vii, viii, ix, 103, 104, 105, 107, 108, 110,

116, 118time, 50Tokyo, 114

topological, 114

topology, 117

transformations, 1, 109

translation, 90transparent, 69

trees, 107

U

unification, 107

USSR, 108, 118

V

values, 107, 108, 115, 116variable, vii, viii, ix, 1, 25, 76, 82, 91, 104, 105, 106,

107, 109, 111, 112, 114, 116, 117

vector, 24, 90

Y

yield, 7, 19, 29, 35, 39, 90, 91

meskhi a. measure of non-compactness for integral operators in weighted lebesgue spaces (nova,...

Documents