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ON SOME RESTRICTED INEQUALITIES FOR THE ITERATED HARDY-TYPE OPERATOR

INVOLVING SUPREMA AND THEIR APPLICATIONS

RZA MUSTAFAYEV, NEVIN BILGICLI, AND MERVE YILMAZ

Abstract. In this paper we characterize the inequality

(

∫ ∞

0

(

∫ x

0

[

Tu,b f ∗(t)]r

dt

)

q

r

w(x)dx

)1q

≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p dτ

)mp

v(x)dx

)1m

for 1 < m < p ≤ r < q <∞ or 1 < m ≤ r < minp,q <∞, where w and v are weight functions on (0,∞). The inequality

is required to hold with some positive constant C for all measurable functions defined on measure space (Rn,dx). Here

f ∗ is the non-increasing rearrangement of a measurable function f defined on Rn and Tu,b is the iterated Hardy-type

operator involving suprema, whish is defined for a measurable non-negative function f on (0,∞) by

(Tu,bg)(t) := supt≤τ<∞

u(τ)

B(τ)

∫ τ

0

g(s)b(s)ds, t ∈ (0,∞),

where u and b are two weight functions on (0,∞) such that u is continuous on (0,∞) and the function B(t) :=∫ t

0b(s)ds

satisfies 0 < B(t) <∞ for every t ∈ (0,∞).

At the end of the paper, as an application of obtained results, we calculate the norm of the generalized maximal

operator Mφ,Λα(b), defined with 0 < α <∞ and functions b, φ : (0,∞)→ (0,∞) for all measurable functions f on Rn by

Mφ,Λα(b) f (x) := supQ∋x

‖ fχQ‖Λα(b)

φ(|Q|), x ∈ Rn,

from GΓ(p1,m1,v) into GΓ(p2,m2,w). Here Λα(b) and GΓ(p,m,w) are the classical and generalized Lorentz spaces,

defined as a set of all measurable functions f defined on Rn for which

‖ f ‖Λα(b) =

(

∫ ∞

0

[ f ∗(s)]αb(s)ds

)1α

<∞ and ‖ f ‖GΓ(p,m,w) =

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p dτ

)mp

v(x)dx

)1m

<∞,

respectively.

1. Introduction

Let (R,µ) be a σ-finite non-atomic measure space. Denote by M(R) the set of all µ-measurable functions on R

andM0(R) the class of functions inM(R) that are finite µ - a.e. on R. The symbolM+(R) stands for the collection

of all f ∈M(R) which are non-negative on R.

The non-increasing rearrangement f ∗ of f ∈M0(R) is given by

f ∗(t) = inf

λ ≥ 0 : µ(x ∈ R : | f (x)| > λ) ≤ t

, t ∈ [0,µ(R)).

The maximal non-increasing rearrangement of f is defined as follows

(1.1) f ∗∗(t) :=1

t

∫ t

0

f ∗(τ)dτ, t ∈ (0,µ(R)).

Most of the functions which we shall deal with will be defined on Rn or (0,∞). In this case, (R,µ) is Rn or (0,∞)

endowed with the n-dimensional Lebesgue measure or the one-dimensional Lebesgue measure, respectively. We

shall write justM+ instead ofM+(0,∞).

Let Ω be any measurable subset of Rn, n ≥ 1. The family of all weight functions (also called just weights) on Ω,

that is, locally integrable non-negative functions on Ω, denoted byW(Ω).

2010 Mathematics Subject Classification. 42B25, 42B35.

Key words and phrases. generalized maximal functions, classical and generalized Lorentz spaces, iterated Hardy inequalities involving

suprema, weights.

1

2 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

For p ∈ (0,∞] and w ∈M+(Ω), we define the functional ‖ · ‖p,w,Ω onM(Ω) by

‖ f ‖p,w,Ω :=

(∫

Ω| f (x)|pw(x)dx

)1/pif p <∞,

esssupΩ | f (x)|w(x) if p =∞.

If, in addition, w ∈W(Ω), then the weighted Lebesgue space Lp(w,Ω) is given by

Lp(w,Ω) = f ∈M(Ω) : ‖ f ‖p,w,Ω <∞

and it is equipped with the quasi-norm ‖ · ‖p,w,Ω.

When w ≡ 1 on Ω, we write simply Lp(Ω) and ‖ · ‖p,Ω instead of Lp(w,Ω) and ‖ · ‖p,w,Ω, respectively.

Quite many familiar function spaces can be defined using the non-increasing rearrangement of a function. One

of the most important classes of such spaces are the so-called classical Lorentz spaces.

Let p ∈ (0,∞) and w ∈W(0,µ(R)). Then the classical Lorentz spaces Λp(w) and Γp(w) consist of all functions

f ∈M(R) for which

‖ f ‖Λp(w) :=

(

∫ µ(R)

0

[ f ∗(s)]pw(s)ds

)1p

<∞ and ‖ f ‖Γp(w) :=

(

∫ µ(R)

0

[ f ∗∗(s)]pw(s)ds

)1p

<∞,

respectively. For more information about the Lorentz Λ and Γ see e.g. [3] and the references therein.

The study of particular problems in the regularity theory of PDE’s led to the definition of spaces involving

inner integral means involving powers of the non-increasing rearrangements of functions. The generalized Lorentz

GΓ(p,m,v)(R,µ) space (denoted simply by GΓ(p,m,v)), introduced and studied in [12] and [13], is defined as the

collection of all g ∈M(R) such that

‖g‖GΓ(p,m,v) =

(

∫ µ(R)

0

(

∫ x

0

[g∗(τ)]p dτ

)mp

v(x)dx

)1m

<∞,

where m, p ∈ (0,∞), v ∈W(0,µ(R)).

The spaces GΓ(p,m,v) cover several types of important function spaces and have plenty of applications. For

example, when µ(R) = ∞, p = 1, m > 1 and v(t) = t−mw(t), t ∈ (0,∞), where w is another weight on (0,∞), then

GΓ(p,m,v) reduces the spaces Γm(w). Another important example is obtained when µ(R) = 1, m = 1, p ∈ (1,∞) and

v(t) = t−1( log(2/t))−1/p

for t ∈ (0,1). In this case the space GΓ(p,m,v) coincides with the small Lebesgue space,

which was originally studied by Fiorenza in [9]. In the same paper it was proved that this space is the associate

space of the grand Lebesgue space introduced in [30] in connection with integrability properties of Jacobians.

Subsequently, Fiorenza and Karadzhov in [11] derived an equivalent form of the norm in the small Lebesgue space

written in the form of the norm in the GΓ(p,m,v) space with the above mentioned parameters and weight. Note

that the condition∫

(0,1)tm/pv(t)dt <∞ ensures GΓ(p,m,v) to be a quasi-Banach function space when R ⊂ Rn with

µ(R) = 1 (cf. [10]). Recently, the connection of the GΓ(p,m,v) space with some well-known function spaces

have been studied in [1]. In the present paper we take (Rn,dx) as underlying measure space and use the notation

GΓ(p,m,v) for GΓ(p,m,v)(Rn,dx).

The study on maximal operators occupies an important place in harmonic analysis. These significant non-linear

operators, whose behavior are very informative in particular in differentiation theory, provided the understanding

and the inspiration for the development of the general class of singular and potential operators (see, for instance,

[14, 27–29, 53, 54, 56]).

The main example is the Hardy-Littlewood maximal function which is defined for locally integrable functions f

on Rn by

(M f )(x) := supQ∋x

1

|Q|

∫

Q

| f (y)|dy, x ∈ Rn,

where the supremum is taken over all cubes Q containing x. By a cube, we mean an open cube with sides parallel

to the coordinate axes.

Another important example is the fractional maximal operator, Mγ, γ ∈ (0,n), defined for locally integrable

functions f on Rn by

(Mγ f )(x) := supQ∋x

|Q|γ/n−1

∫

Q

| f (y)|dy, x ∈ Rn.

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 3

One more example is the fractional maximal operator Ms,γ,A defined in [7] for all measurable functions f on Rn

by

(Ms,γ,A f )(x) := supQ∋x

‖ fχQ‖s

‖χQ‖sn/(n−γ),A

, x ∈ Rn.

Here s ∈ (0,∞), γ ∈ [0,n), A = (A0,A∞) ∈ R2 and

ℓA(t) := (1+ | log t|)A0χ[0,1](t)+ (1+ | log t|)A∞χ[1,∞)(t), t ∈ (0,∞).

Recall that the following equivalency holds:

(Ms,γ,A f )(x) ≈ supQ∋x

‖ fχQ‖s

|Q|(n−γ)/(sn)ℓA(|Q|), x ∈ Rn.

Hence, if s = 1, γ = 0 and A = (0,0), then Ms,γ,A is equivalent to M. If s = 1, γ ∈ (0,n) and A = (0,0), then Ms,γ,A

is equivalent to Mγ. Moreover, if s = 1, γ ∈ [0,n) and A ∈ R2, then Ms,γ,A is the fractional maximal operator which

corresponds to potentials with logarithmic smoothness treated in [39, 40]. In particular, if γ = 0, then M1,γ,A is the

maximal operator of purely logarithmic order.

Given p and q, 0 < p, q <∞, let Mp,q denote the maximal operator associated to the Lorentz Lp,q spaces defined

for all measurable function f on Rn by

Mp,q f (x) := supQ∋x

‖ fχQ‖p,q

‖χQ‖p,q,

where ‖ · ‖p,q is the usual Lorentz norm

‖ f ‖p,q :=

(

∫ ∞

0

[

τ1/p f ∗(τ)]q dτ

τ

)1/q

.

This operator was introduced by Stein in [52] in order to obtain certain endpoint results in differentiation theory.

The operator Mp,q have been also considered by other authors, for instance see [2, 33, 34, 38, 42].

Let 0 < α <∞, b ∈W(0,∞) and φ : (0,∞)→ (0,∞). Recall the definition of the generalized maximal function

introduced in [36] and denoted for all measurable function f on Rn by

(1.2) Mφ,Λα(b) f (x) := supQ∋x

‖ fχQ‖Λα(b)

φ(|Q|), x ∈ Rn.

Obviously, Mφ,Λα(b) = M, where M is the Hardy-Littlewood maximal operator, when α = 1, b ≡ 1 and φ(t) = t

(t > 0). Note that Mφ,Λα(b) = Mγ, where Mγ is the fractional maximal operator, when α = 1, b ≡ 1 and φ(t) = t1−γ/n

(t > 0) with 0 < γ < n. Moreover, Mφ,Λα(b) ≈Ms,γ,A, when α= s, b ≡ 1 and φ(t) = t(n−γ)/(sn)ℓA(t) (t > 0) with 0< γ < n

and A = (A0,A∞) ∈ R2. It is worth also to mention that Mφ,Λα(b) = Mp,q, when α = q, b(t) = tq/p−1 and φ(t) = t1/p

(t > 0).

The boundedness of Mφ,Λα(b) between classical Lorentz spaces Λ was completely characterized in [36]. In view

of importance of generalized Lorentz spaces in our opinion it will be interesting to obtain a characterization of the

boundedness of this maximal function between Lorentz GΓ spaces.

The iterated Hardy-type operator involving suprema Tu,b is defined for non-negative measurable function g on

the interval (0,∞) by

(Tu,bg)(t) := supt≤τ<∞

u(τ)

B(τ)

∫ τ

0

g(y)b(y)dy, t ∈ (0,∞),

where u and b are two weight functions on (0,∞) such that u is continuous on (0,∞) and the function B(t) :=∫ t

0b(s)ds satisfies 0 < B(t) <∞ for every t ∈ (0,∞). Such operators have been found indispensable in the search

for optimal pairs of rearrangement-invariant norms for which a Sobolev-type inequality holds (cf. [31]). They

constitute a very useful tool for characterization of the associate norm of an operator-induced norm, which naturally

appears as an optimal domain norm in a Sobolev embedding (cf. [41,43]). Supremum operators are also very useful

in limiting interpolation theory as can be seen from their appearance for example in [4, 5, 8, 48].

The aim of the paper is to characterize the following restricted inequality for Tu,b:

(1.3)

(

∫ ∞

0

(

∫ x

0

[

Tu,b f ∗(t)]r

dt

)

q

r

w(x)dx

)1q

≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p dτ

)mp

v(x)dx

)1m

.

Here m, p, q, r are positive real numbers and w, v are weight functions on (0,∞).

4 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

A function φ : (0,∞)→ (0,∞) is said to satisfy the ∆2 - condition, denored φ ∈ ∆2, if for some C > 0

φ(2t) ≤Cφ(t) for all 0 < t <∞.

A function φ : (0,∞)→ (0,∞) is said to be quasi-increasing, if for some C > 0

φ(t1) ≤Cφ(t2),

whenever 0 < t1 ≤ t2 <∞.

A function φ : (0,∞)→ (0,∞) is said to satisfy the Qr-condition, 0 < r <∞, denoted φ ∈ Qr(0,∞), if for some

constant C > 0

φ

( n∑

i=1

ti

)

≤C

( n∑

i=1

φ(ti)r)1/r

,

for every finite set of non-negative real numbers t1, . . . , tn.

Motivation for studying inequality (1.3) comes directly from the following eqiuvalency statement.

Theorem 1.1. Let 0 < p1, p2 <∞, 0 < m1, m2 <∞, 0 < α ≤ r <∞ and v, w ∈W(0,∞). Assume that φ ∈ Qr(0,∞)

is a quasi-increasing function. Suppose that b ∈W(0,∞) is such that 0 < B(t) <∞ for all t > 0, B ∈ ∆2, B(∞) =∞

and B(t)/tα/r is quasi-increasing. Then the inequality

(1.4)

(

∫ ∞

0

(

∫ x

0

[(

Mφ,Λα(b) f)∗

(t)]p2 dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p1 dτ

)

m1p1

v(x)dx

)1

m1

holds for all f ∈M(Rn) if and only if the inequality

(

∫ ∞

0

(

∫ x

0

[

TB/φα,bh∗(t)]

p2α dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[h∗(τ)]p1α dτ

)

m1p1

v(x)dx

)1

m1

holds for all h ∈M(Rn).

The method used for solution of inequalty (1.3) is based on the combination of the duality techniques with the

formula

supg:

∫ x

0g≤

∫ x

0f

∫ ∞

0

g(x)w(x)dx =

∫ ∞

0

f (x)

(

supt≥x

w(t)

)

dx

from [50], which holds for f , w ∈M+(0,∞). On the other hand, it uses estimates of optimal constants in iterated

Hardy-type inequalities, ”gluing” lemmas, which allows to reduce the problem to using of the boundedness of

weighted iterated Hardy-type operators involving suprema from weighted Lebesgue spaces into weighted Cesaro

function spaces. Detailed information on materials that are used in the proofs of the main results is given in the

following section.

It should be noted that the method developed in the present paper allows to solve inequality (1.3) in the case

0 < m ≤ 1, 0 < p < ∞ or 1 < m <∞, 0 < p ≤ 1, as well. However, we are able to solve the inequality under the

restrictions 1 < m < p ≤ r < q < ∞ or 1 < m ≤ r < minp,q < ∞. For the remaining values of parameters the

conditions that characterize the weighted iterated Hardy-type inequalities contains more complicated expressions

and the approach used in our paper needs an improvement.

Using duality principle puts restriction r ≤ q on parameters. But for r = q inequality (1.3) is a special case of the

inequality

(1.5)

(

∫ ∞

0

[

Tu,b f ∗(x)]q

w(x)dx

)1q

≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p dτ

)mp

v(x)dx

)1m

,

which we are going to investigate in our forthcoming studies.

Throughout the paper, we always denote by C a positive constant, which is independent of main parameters but it

may vary from line to line. However a constant with subscript such as C1 does not change in different occurrences.

By a . b, we mean that a ≤ λb, where λ > 0 depends on inessential parameters. If a . b and b . a, we write a ≈ b

and say that a and b are equivalent.

As usual, we put 0 ·∞ = 0, ∞/∞ = 0 and 0/0 = 0. If p ∈ [1,+∞], we define p′ by 1/p+1/p′ = 1.

The paper is organized as follows. We start with formulations of background material in Section 2. In Section 3

we present solution of the restricted inequality. Finally, in Section 4 we calculate the norm of generalized maximal

function between Lorentz GΓ spaces.

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 5

2. Background material

In this section we collect background material that will be used in the proofs of the main theorems.

We begin with the following characterization of the norm of the associate space of GΓ(p,m,v) given in [26].

Recall that the associate space GΓ(p,m,v)′ of GΓ(p,m,v) is defined as the collection of all functions g ∈M(Rn)

such that

‖g‖GΓ(p,m,v)′ = sup‖ f ‖GΓ(p,m,v)≤1

∫ ∞

0

f ∗(t)g∗(t)dt <∞.

As it is mentioned in [26], it is reasonable to adopt a general assumption that p, m and v are such that

(2.1)

∫ t

0

v(s)smp ds+

∫ ∞

t

v(s)ds <∞, t ∈ (0,∞),

because if this requirement is not satisfied, then GΓ(p,m,v) = 0.

Under the assumption (2.1), we denote

(2.2) v0(t) := tmp−1

∫ t

0

v(s)smp ds

∫ ∞

t

v(s)ds, t ∈ (0,∞),

and

(2.3) v1(t) :=

∫ t

0

v(s)smp ds+ t

mp

∫ ∞

t

v(s)ds, t ∈ (0,∞).

Moreover, we assume that a weight v is non-degenerate (with respect to the power function tm/p), that is,

(2.4)

∫ 1

0

v(s)ds =

∫ ∞

1

v(s)smp ds =∞.

We denote the set of all weight functions satisfying conditions (2.1) and (2.4) byWm,p(0,∞).

Theorem 2.1. [26, Theorem 1.1] Assume that 1 < m, p <∞ and v ∈Wm,p(0,∞). Then

‖g‖GΓ(p,m,v)′ ≈

(

∫ ∞

0

(

∫ ∞

t

g∗∗(s)p′ ds

)m′

p′ tm′

p′ v0(t)

v1(t)m′+1dt

)1

m′

.

We recall the following well-known duality principle in weighted Lebesgue spaces.

Theorem 2.2. Let p > 1, f ∈M+(0,∞) and w ∈W(0,∞). Then

(

∫ ∞

0

f (t)pw(t)dt

)1p

= suph∈M+

∫ ∞

0f (t)h(t)dt

(

∫ ∞

0h(t)p′w(t)1−p′ dt

)1p′

.

We will use the following statement.

Theorem 2.3. [50, Theorem 2.1] Suppose f , w ∈M+. Then

supg:

∫ x

0g≤

∫ x

0f

∫ ∞

0

g(x)w(x)dx =

∫ ∞

0

f (x)

(

supt≥x

w(t)

)

dx.

We will apply the following ”gluing” lemma.

Lemma 2.4. [23, Lemma 2.7] Let α and β be positive numbers. Suppose that g, h ∈ M+ and a ∈ W(0,∞) is

non-decreasing. Then

esssupx∈(0,∞)

(

∫ ∞

0

(

a(x)

a(x)+a(t)

)β

g(t)dt

)1β(

∫ ∞

0

(

a(t)

a(x)+a(t)

)α

h(t)dt

)1α

≈ esssupx∈(0,∞)

(

∫ x

0

g(t)dt

)1β(

∫ ∞

x

h(t)dt

)1α

+ esssupx∈(0,∞)

(

∫ ∞

x

a(t)−βg(t)dt

)1β(

∫ x

0

a(t)αh(t)dt

)1α

.

We recall the following ”an integration by parts” formula.

6 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

Theorem 2.5. [37, Theorem 2.1] Let α > 0. Let g be a non-negative function on (0,∞) such that 0 <∫ t

0g <∞ for

all t ∈ (0,∞) and let f be a non-negative non-increasing right-continuous function on (0,∞). Then

A1 :=

∫ ∞

0

(

∫ t

0

g

)α

g(t)[ f (t)− limt→+∞

f (t)]dt <∞ ⇐⇒ A2 :=

∫

(0,∞)

(

∫ t

0

g

)α+1

d[− f (t)] <∞.

Moreover, A1 ≈ A2.

Investigation of weighted iterated Hardy-type inequalities started with studying of the inequality

(2.5)

(

∫ ∞

0

(

∫ t

0

(

∫ ∞

s

h(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

≤C

(

∫ ∞

0

h(t)pv(t)dt

)1p

, h ∈M+.

Note that inequality (2.5) have been considered in the case m = 1 in [6] (see also [15]), where the result was

presented without proof, and in the case p = 1 in [16] and [51], where the special type of weight function v was

considered. Recall that the inequality has been completely characterized in [19] and [20] in the case 0 < m <

∞, 0 < q ≤ ∞, 1 ≤ p < ∞ by using discretization and anti-discretization methods. Another approach to get the

characterization of inequalities (2.5) was presented in [44]. The characterization of the inequality can be reduced

to the characterization of the weighted Hardy inequality on the cones of non-increasing functions (see, [21] and

[22]). Different approach to solve iterated Hardy-type inequalities has been given in [35].

As it was mentioned in [21] the characterization of ”dual” inequality

(2.6)

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

h(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

≤C

(

∫ ∞

0

h(t)pv(t)dt

)1p

, h ∈M+.

can be easily obtained from the solutions of inequality (2.5), which was presented in [17].

Theorem 2.6. [17, Theorem 2.9, (a) and (c)] Let p, q, m ∈ (1,∞) and u, w, v be weights on (0,∞). Assume that the

following non-degeneracy conditions are satisfied:

• u is strictly positive,∫ ∞

tu(s)ds <∞ for all t ∈ (0,∞),

∫ ∞

0u(s)ds =∞,

•∫ t

0w(s)ds <∞,

∫ ∞

tw(s)

(

∫ ∞

su(y)dy

)

q

m

ds <∞ for all t ∈ (0,∞),

•∫ 1

0w(s)

(

∫ ∞

su(y)dy

)

q

m

ds =∞,∫ ∞

1w(s)ds =∞.

Let

C = suph∈M+

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0h(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

(

∫ ∞

0h(t)pv(t)dt

)1p

(a) If p ≤ m and p ≤ q, then C ≈ D1+D2, where

D1 = supt∈(0,∞)

(

∫ t

0

w(s)ds

)1q(

∫ ∞

t

u(s)ds

)1m(

∫ t

0

v(τ)1−p′ dτ

)1p′

and

D2 = supt∈(0,∞)

(

∫ ∞

t

(

∫ ∞

s

u(y)dy

)

q

m

w(s)ds

)1q(

∫ t

0

v(s)1−p′ ds

)1p′

.

(b) If m < p and p ≤ q, then C ≈ D2+D3, where

D3 = supt∈(0,∞)

(

∫ t

0

w(s)ds

)1q(

∫ ∞

t

(

∫ ∞

s

u(y)dy

)

p

p−m(

∫ s

0

v(τ)1−p′ dτ

)

p(m−1)p−m

v(s)1−p′ ds

)

p−m

pm

.

Another pair of ”dual” weighted iterated Hardy-type inequalities are

(2.7)

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

h(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

≤C

(

∫ ∞

0

h(t)pv(t)dt

)1p

, h ∈M+

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 7

and

(2.8)

(

∫ ∞

0

(

∫ t

0

(

∫ s

0

h(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

≤C

(

∫ ∞

0

h(t)pv(t)dt

)1p

, h ∈M+.

Both of them were characterized in [21] by so-called ”flipped” conditions. The ”classical” conditions ensuring the

validity of (2.7) was recently presented in [32].

Theorem 2.7. [32, Theorem 1.1, (a) and (c)] Let p, q, m ∈ (1,∞) and u, w, v be weights such that the pair (u,w) is

admissible with respect to (m.q), that is,

0 <

∫ t

0

(

∫ t

s

u(y)dy

)

q

m

w(s)ds <∞, t ∈ (0,∞).

Let

C = suph∈M+

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

sh(y)dy

)m

u(s)ds

)

q

m

w(t)dt

)1q

(

∫ ∞

0h(t)pv(t)dt

)1p

(a) If p ≤ m and p ≤ q, then C ≈ E1, where

E1 = supt∈(0,∞)

(

∫ t

0

w(s)

(

∫ t

s

u(y)dy

)

q

m

ds

)1q(

∫ ∞

t

v(s)1−p′ ds

)1p′

.

(b) If m < p and p ≤ q, then C ≈ E1+E2, where

E2 = supt∈(0,∞)

(

∫ t

0

w(s)ds

)1q(

∫ ∞

t

(

∫ s

t

u(y)dy

)m

p−m

u(s)

(

∫ ∞

s

v(τ)1−p′ dτ

)

m(p−1)p−m

ds

)

p−m

pm

.

We need solutions of inequalities

(

∫ ∞

t

(

∫ x

t

(

sups≤τ

u(τ)

(

∫ τ

t

f (y)dy

))

a(s)ds

)q

w(x)dx

)1q

≤ C

(

∫ ∞

t

f (s)pv(s)ds

)1p

, f ∈M+(t,∞)

and(

∫ ∞

t

(

∫ y

t

(

sups≤τ

u(τ)

(

∫ ∞

τ

f (z)dz

))

a(s)ds

)q

w(y)dy

)1q

≤ C

(

∫ ∞

t

f (s)pv(s)ds

)1p

, f ∈M+(t,∞)

where t is a given pont in [0,∞) and 1 < p, q <∞ and u ∈W(t,∞)∩C(t,∞) and a,v, w ∈W(t,∞).

The proofs of the following two statements are straightforward and capacious, but for the convenience of the

reader we give the complete proof of both of them.

Theorem 2.8. Let 1 < p,q <∞. Given t ≥ 0 assume that u ∈W(t,∞)∩C(t,∞) and a,v, w ∈W(t,∞). Moreover,

assume that 0 <∫ x

0v(τ)1−p′ dτ <∞, x > t.

• If p ≤ q, then

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ x

t

(

sups≤τ u(τ)

(

∫ τ

tf (y)dy

))

a(s)ds

)q

w(x)dx

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈ supy∈(t,∞)

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z)dz

)1q

+ supy∈(t,∞)

(

∫ y

t

v(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)1q

+ supx∈(t,∞)

(

∫

[x,∞)

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)1q

8 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+ supx∈(t,∞)

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x

w(z)dz

)1q

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

.

• If q < p, then

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ x

t

(

sups≤τ u(τ)

(

∫ τ

tf (y)dy

))

a(s)ds

)q

w(x)dx

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈

(

∫ ∞

t

(

∫ y

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)

pp−q

dy

)

p−qpq

+

(

∫ ∞

t

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

+

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

.

Proof. The statement was formulated in [37, Theorem 3.1] for t = 0. Assume that t > 0. The proof uses multiple

changes of variables of the type x+ t = y several times, which we will not specify at any step exactly.

It is easy to see that, given a point t ∈ (0,∞), the inequality

(

∫ ∞

t

(

∫ x

t

(

sups≤τ

u(τ)

(

∫ τ

t

f (y)dy

))

a(s)ds

)q

w(x)dx

)1q

≤C

(

∫ ∞

t

f (s)pv(s)ds

)1p

, f ∈M+(t,∞)(2.9)

is equivalent to the inequality

(2.10)

(

∫ ∞

0

(

∫ x

0

(

sups≤τ

ut(τ)

(

∫ τ

0

f (y)dy

))

at(s)ds

)q

wt(x)dx

)1q

≤C

(

∫ ∞

0

f (s)pvt(s)ds

)1p

, f ∈M+(0,∞),

where the weight functions ut, wt, vt and at are defined for x > 0 as follows:

ut(x) := u(x+ t), wt(x) := w(x+ t), vt(x) := v(x+ t), at(x) := a(x+ t).

Indeed: Applying changes of variables, we get that∫ ∞

t

(

∫ x

t

(

sups≤τ

u(τ)

(

∫ τ

t

f (z)dz

))

a(s)ds

)q

w(x)dx

=

∫ ∞

t

(

∫ x

t

(

sups≤τ

u(τ)

(

∫ τ−t

0

f (z+ t)dz

))

a(s)ds

)q

w(x)dx

=

∫ ∞

t

(

∫ x

t

(

sups−t≤τ−t

u(τ)

(

∫ τ−t

0

f (z+ t)dz

))

a(s)ds

)q

w(x)dx

=

∫ ∞

t

(

∫ x

t

(

sups−t≤τ

u(τ+ t)

(

∫ τ

0

f (z+ t)dz

))

a(s)ds

)q

w(x)dx

=

∫ ∞

t

(

∫ x−t

0

(

sups≤τ

u(τ+ t)

(

∫ τ

0

f (z+ t)dz

))

a(s+ t)ds

)q

w(x)dx

=

∫ ∞

0

(

∫ x

0

(

sups≤τ

u(τ+ t)

(

∫ τ

0

f (z+ t)dz

))

a(s+ t)ds

)q

w(x+ t)dx

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 9

=

∫ ∞

0

(

∫ x

0

(

sups≤τ

ut(τ)

(

∫ τ

0

f (z+ t)dz

))

at(s)ds

)q

wt(x)dx

and∫ ∞

t

f (s)pv(s)ds =

∫ ∞

0

f (s+ t)pv(s+ t)ds =

∫ ∞

0

f (s+ t)pvt(s)ds.

Hence inequality (2.9) can be rewritten as follows:

(

∫ ∞

0

(

∫ y

0

(

sups≤τ

ut(τ)

(

∫ ∞

τ

f (z+ t)dz

))

at(s)ds

)q

wt(y)dy

)1q

≤C

(

∫ ∞

0

f (s+ t)pvt(s)ds

)1p

, f ∈M+(t,∞).

It remains note that the latter is equivalent to inequality (2.10).

Let p ≤ q. By [37, Theorem 3.1], we have for any t ∈ (0,∞) that

supf∈M+(0,∞)

(

∫ ∞

0

(

∫ x

0

(

sups≤τ ut(τ)

(

∫ τ

0f (y)dy

))

at(s)ds

)q

wt(x)dx

)1q

(

∫ ∞

0f (s)pvt(s)ds

)1p

≈ supy∈(0,∞)

(

∫ y

0

vt(x)1−p′(

∫ y

x

(

sups≤τ

ut(τ)

)

at(s)ds

)p′

dx

)1p′

(

∫ ∞

y

wt(z)dz

)1q

+ supy∈(0,∞)

(

∫ y

0

vt(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)

)

at(s)ds

)q

wt(z)dz

)1q

+ supx∈(0,∞)

(

∫

[x,∞)

d

(

− supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

)))1p′

(

∫ x

0

At(z)qwt(z)dz

)1q

+ supx∈(0,∞)

(

∫

(0,x]

At(y)p′ d

(

− supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

)))1p′

(

∫ ∞

x

wt(z)dz

)1q

+

(

∫ ∞

0

At(z)qwt(z)dz

)1q

limx→∞

(

supx≤τ

ut(τ)

(

∫ τ

0

vt(s)1−p′ ds

)1p′)

.

Since

supy∈(0,∞)

(

∫ y

0

vt(x)1−p′(

∫ y

x

(

sups≤τ

ut(τ)

)

at(s)ds

)p′

dx

)1p′

(

∫ ∞

y

wt(z)dz

)1q

= supy∈(0,∞)

(

∫ y

0

v(x+ t)1−p′(

∫ y

x

(

sups≤τ

u(τ+ t)

)

a(s+ t)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y

0

v(x+ t)1−p′(

∫ y

x

(

sups+t≤τ

u(τ)

)

a(s+ t)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y

0

v(x+ t)1−p′(

∫ y+t

x+t

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y+t

t

v(x)1−p′(

∫ y+t

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)1p′

(

∫ ∞

y+t

w(z)dz

)1q

= supy∈(t,∞)

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z)dz

)1q

,

supy∈(0,∞)

(

∫ y

0

vt(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)

)

at(s)ds

)q

wt(z)dz

)1q

= supy∈(0,∞)

(

∫ y

0

v(x+ t)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ+ t)

)

a(s+ t)ds

)q

w(z+ t)dz

)1q

10 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

= supy∈(0,∞)

(

∫ y+t

t

v(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z+t

y+t

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y+t

t

v(x)1−p′ dx

)1p′

(

∫ ∞

y+t

(

∫ z

y+t

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)1q

= supy∈(t,∞)

(

∫ y

t

v(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)1q

,

supx∈(0,∞)

(

∫

[x,∞)

d

(

− supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

)))1p′

(

∫ x

0

At(z)qwt(z)dz

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− supy≤τ

u(τ+ t)p′(

∫ τ

0

v(s+ t)1−p′ ds

)))1p′

(

∫ x

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− supy≤τ

u(τ+ t)p′(

∫ τ+t

t

v(s)1−p′ ds

)))1p′

(

∫ x

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− supy+t≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ x

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

[x+t,∞)

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)1q

= supx∈(t,∞)

(

∫

[x,∞)

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)1q

,

supx∈(0,∞)

(

∫

(0,x]

At(y)p′ d

(

− supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

)))1p′

(

∫ ∞

x

wt(z)dz

)1q

= supx∈(0,∞)

(

∫

(0,x]

(

∫ y+t

t

a

)p′

d

(

− supy≤τ

u(τ+ t)p′(

∫ τ

0

v(s+ t)1−p′ ds

)))1p′

(

∫ ∞

x

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

(0,x]

(

∫ y+t

t

a

)p′

d

(

− supy≤τ

u(τ+ t)p′(

∫ τ+t

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

(0,x]

(

∫ y+t

t

a

)p′

d

(

− supy+t≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

(t,x+t]

(

∫ y

t

a

)p′

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x+t

w(z)dz

)1q

= supx∈(t,∞)

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x

w(z)dz

)1q

,

(

∫ ∞

0

At(z)qwt(z)dz

)1q

limx→∞

(

supx≤τ

ut(τ)

(

∫ τ

0

vt(s)1−p′ ds

)1p′)

=

(

∫ ∞

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)1q

limx→∞

(

supx≤τ

u(τ+ t)

(

∫ τ

0

v(s+ t)1−p′ ds

)1p′)

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ+ t)

(

∫ τ+t

t

v(s)1−p′ ds

)1p′)

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx+t≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 11

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

,

we arrive at

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ x

t

(

sups≤τ u(τ)

(

∫ τ

tf (y)dy

))

a(s)ds

)q

w(x)dx

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈ supy∈(t,∞)

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)1p′

(

∫ ∞

y

w(z)dz

)1q

+ supy∈(t,∞)

(

∫ y

t

v(x)1−p′ dx

)1p′

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)1q

+ supx∈(t,∞)

(

∫

[x,∞)

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)1q

+ supx∈(t,∞)

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

− supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

)))1p′

(

∫ ∞

x

w(z)dz

)1q

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

.

Let q < p. By [37, Theorem 3.1], we have for any t ∈ (0,∞) that

supf∈M+(0,∞)

(

∫ ∞

0

(

∫ x

0

(

sups≤τ ut(τ)

(

∫ τ

0f (y)dy

))

at(s)ds

)q

wt(x)dx

)1q

(

∫ ∞

0f (s)pvt(s)ds

)1p

≈

(

∫ ∞

0

(

∫ y

0

vt(x)1−p′ dx

)

p(q−1)p−q

vt(y)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)

)

at(s)ds

)q

wt(z)dz

)

p

p−q

dy

)

p−q

pq

+

(

∫ ∞

0

(

∫ y

0

vt(x)1−p′(

∫ y

x

(

sups≤τ

ut(τ)

)

at(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

wt(z)dz

)

q

p−q

wt(y)dy

)

p−q

pq

+

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x

0

At(z)qwt(z)dz

)

q

p−q

At(x)qwt(x)dx

)

p−q

pq

+

(

∫ ∞

0

(

∫

(0,x]

At(y)p′ d

(

−

(

supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

wt(z)dz

)

q

p−q

wt(x)dx

)

p−q

pq

+

(

∫ ∞

0

At(z)qwt(z)dz

)1q

limx→∞

(

supx≤τ

ut(τ)

(

∫ τ

0

vt(s)1−p′ ds

)1p′)

.

Since

(

∫ ∞

0

(

∫ y

0

vt(x)1−p′ dx

)

p(q−1)p−q

vt(y)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)

)

at(s)ds

)q

wt(z)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

v(x+ t)1−p′ dx

)

p(q−1)p−q

v(y+ t)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ+ t)

)

a(s+ t)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y+ t)1−p′(

∫ ∞

y

(

∫ z

y

(

sups+t≤τ

u(τ)

)

a(s+ t)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y+ t)1−p′(

∫ ∞

y

(

∫ z+t

y+t

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

12 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

=

(

∫ ∞

0

(

∫ y+t

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y+ t)1−p′(

∫ ∞

y+t

(

∫ z

y+t

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

t

(

∫ y

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)

pp−q

dy

)

p−q

pq

,

(

∫ ∞

0

(

∫ y

0

vt(x)1−p′(

∫ y

x

(

sups≤τ

ut(τ)

)

at(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

wt(z)dz

)

q

p−q

wt(y)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

v(x+ t)1−p′(

∫ y

x

(

sups≤τ

u(τ+ t)

)

a(s+ t)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z+ t)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

v(x+ t)1−p′(

∫ y

x

(

sups+t≤τ

u(τ)

)

a(s+ t)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y+t

w(z)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

v(x+ t)1−p′(

∫ y+t

x+t

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y+t

w(z)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

v(x)1−p′(

∫ y+t

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y+t

w(z)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

t

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

,

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x

0

At(z)qwt(z)dz

)

q

p−q

At(x)qwt(x)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τ

u(τ+ t)p′(

∫ τ

0

v(s+ t)1−p′ ds

))))

q(p−1)p−q

(

∫ x

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τ

u(τ+ t)p′(

∫ τ+t

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy+t≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x+t,∞)

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

,

(

∫ ∞

0

(

∫

(0,x]

At(y)p′ d

(

−

(

supy≤τ

ut(τ)p′(

∫ τ

0

vt(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

wt(z)dz

)

q

p−q

wt(x)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(0,x]

(

∫ y+t

t

a

)p′

d

(

−

(

supy≤τ

u(τ+ t)p′(

∫ τ

0

v(s+ t)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

w(z+ t)dz

)

q

p−q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(0,x]

(

∫ y+t

t

a

)p′

d

(

−

(

supy+t≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x+t

w(z)dz

)

q

p−q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(t,x+t]

(

∫ y

t

a

)p′

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x+t

w(z)dz

)

qp−q

w(x+ t)dx

)

p−qpq

=

(

∫ ∞

t

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

,

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 13

we arrive at

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ x

t

(

sups≤τ u(τ)

(

∫ τ

tf (y)dy

))

a(s)ds

)q

w(x)dx

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈

(

∫ ∞

t

(

∫ y

t

v(x)1−p′ dx

)

p(q−1)p−q

v(y)1−p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

u(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

+

(

∫ ∞

t

(

∫ y

t

v(x)1−p′(

∫ y

x

(

sups≤τ

u(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

+

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫

(t,x]

(

∫ y

t

a

)p′

d

(

−

(

supy≤τ

u(τ)p′(

∫ τ

t

v(s)1−p′ ds

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

limx→∞

(

supx≤τ

u(τ)

(

∫ τ

t

v(s)1−p′ ds

)1p′)

.

The proof is completed.

Theorem 2.9. Let 1 < p, q <∞. Given t ≥ 0 assume that u ∈W(t,∞)∩C(t,∞) and a,v, w ∈W(t,∞). Moreover,

assume that 0 <∫ ∞

xv(τ)1−p′ dτ <∞, x > t. Denote by

V(x) :=

(

∫ ∞

x

v1−p′)−

2p′

p′+1

v(x)1−p′ , U(x) := u(x)

(

∫ ∞

x

v1−p′)

2p′+1

, x ∈ (0,∞).

• If p ≤ q, then

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ y

t

(

sups≤τ u(τ)

(

∫ ∞

τf (z)dz

))

a(s)ds

)q

w(y)dy

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈ supy∈(t,∞)

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)1p′(

∫ ∞

y

w(x)dx

)1q

+ supy∈(t,∞)

(

∫ y

t

V(x)dx

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)1q

+ supx∈(t,∞)

(

∫

[x,∞)

d

(

− sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ x

t

(

∫ y

t

a

)q

w(y)dy

)1q

+ supx∈(t,∞)

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

− sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

)))1p′

(

∫ ∞

x

w(z)dz

)1q

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τ

U(τ)

(

∫ τ

t

V(y)dy

)1p′)

+

(

∫ ∞

t

v(s)1−p′ ds

)− 1p(p′+1)

(

∫ ∞

t

(

∫ x

t

(

sups≤τ

U(τ)

)

a(s)ds

)q

w(x)dx

)1q

.

14 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

• If q < p, then

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ y

t

(

sups≤τ u(τ)

(

∫ ∞

τf (z)dz

))

a(s)ds

)q

w(y)dy

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈

(

∫ ∞

t

(

∫ y

t

V(x)dx

)

p(q−1)p−q

V(y)

(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

+

(

∫ ∞

t

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

+

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

−

(

sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τU(τ)

(

∫ τ

t

V(y)dy

)1p′)

+

(

∫ ∞

t

v(s)1−p′ ds

)− 1p(p′+1)

(

∫ ∞

t

(

∫ x

t

(

sups≤τU(τ)

)

a(s)ds

)q

w(x)dx

)1q

.

Proof. The statement was formulated in [37, Theorem 3.3] for t = 0. Note that, given a point t ∈ (0,∞), the

inequality

(2.11)

(

∫ ∞

t

(

∫ y

t

(

sups≤τ

u(τ)

(

∫ ∞

τ

f (z)dz

))

a(s)ds

)q

w(y)dy

)1q

≤C

(

∫ ∞

t

f (s)pv(s)ds

)1p

, f ∈M+(t,∞)

is equivalent to the inequality

(2.12)

(

∫ ∞

0

(

∫ y

0

(

sups≤τ

ut(τ)

(

∫ ∞

τ

f (z)dz

))

at(s)ds

)q

wt(y)dy

)1q

≤C

(

∫ ∞

0

f (s)pvt(s)ds

)1p

, f ∈M+(0,∞).

Indeed: Applying changes of variables, we get that∫ ∞

t

(

∫ y

t

(

sups≤τ

u(τ)

(

∫ ∞

τ

f (z)dz

))

a(s)ds

)q

w(y)dy

=

∫ ∞

t

(

∫ y

t

(

sups≤τ

u(τ)

(

∫ ∞

τ−t

f (z+ t)dz

))

a(s)ds

)q

w(y)dy

=

∫ ∞

t

(

∫ y

t

(

sups−t≤τ−t

u(τ)

(

∫ ∞

τ−t

f (z+ t)dz

))

a(s)ds

)q

w(y)dy

=

∫ ∞

t

(

∫ y

t

(

sups−t≤τ

u(τ+ t)

(

∫ ∞

τ

f (z+ t)dz

))

a(s)ds

)q

w(y)dy

=

∫ ∞

t

(

∫ y−t

0

(

sups≤τ

u(τ+ t)

(

∫ ∞

τ

f (z+ t)dz

))

a(s+ t)ds

)q

w(y)dy

=

∫ ∞

0

(

∫ y

0

(

sups≤τ

u(τ+ t)

(

∫ ∞

τ

f (z+ t)dz

))

a(s+ t)ds

)q

w(y+ t)dy

=

∫ ∞

0

(

∫ y

0

(

sups≤τ

ut(τ)

(

∫ ∞

τ

f (z+ t)dz

))

at(s)ds

)q

wt(y)dy

and∫ ∞

t

f (s)pv(s)ds =

∫ ∞

0

f (s+ t)pv(s+ t)ds =

∫ ∞

0

f (s+ t)pvt(s)ds.

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 15

Hence inequality (2.11) can be rewritten as follows:

(

∫ ∞

0

(

∫ y

0

(

sups≤τ

ut(τ)

(

∫ ∞

τ

f (z+ t)dz

))

at(s)ds

)q

wt(y)dy

)1q

≤C

(

∫ ∞

0

f (s+ t)pvt(s)ds

)1p

, f ∈M+(t,∞).

It remains note that the latter is equivalent to inequality (2.12).

Let p ≤ q. By [37, Theorem 3.3], we have for any t ∈ (0,∞) that

supf∈M+(0,∞)

(

∫ ∞

0

(

∫ y

0

(

sups≤τ ut(τ)

(

∫ ∞

τf (z)dz

))

at(s)ds

)q

wt(y)dy

)1q

(

∫ ∞

0f (s)pvt(s)ds

)1p

≈ supy∈(0,∞)

(

∫ y

0

Ψt(x)−p′ψt(x)

(

∫ y

x

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)p′

dx

)1p′(

∫ ∞

y

wt(x)dx

)1q

+ supy∈(0,∞)

(

∫ y

0

Ψt(x)−p′ψt(x)

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(z)dz

)1q

+ supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)))1p′

(

∫ x

0

At(y)q wt(y)dy

)1q

+ supx∈(0,∞)

(

∫

(0,x]

At(s)p′ d

(

− sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)))1p′

(

∫ ∞

x

wt(z)dz

)1q

+

(

∫ ∞

0

At(z)q wt(z)dz

)1q

lims→∞

(

sups≤τ

ut(τ)Ψt(τ)2(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)1p′)

+

(

∫ ∞

0

ψt(z)dz

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(x)dx

)1q

,

where

At(x) :=

∫ x

0

at(z)dz, ψt(x) :=

(

∫ ∞

x

vt(s)1−p′ ds

)−p′

p′+1

vt(x)1−p′ , Ψt(x) :=

(

∫ ∞

x

vt(s)1−p′ ds

)1

p′+1

, x ∈ (0,∞).

Using changes of variables, we get for any x > 0 that

At(x) =

∫ x

0

a(z+ t)dz =

∫ x+t

t

a(z)dz,

ψt(x) =

(

∫ ∞

x

v(s+ t)1−p′ ds

)−p′

p′+1

v(x+ t)1−p′

=

(

∫ ∞

x+t

v(s)1−p′ ds

)−p′

p′+1

v(x+ t)1−p′ ,

and

Ψt(x) =

(

∫ ∞

x+t

v(s)1−p′ ds

)1

p′+1

.

Since

supy≥0

(

∫ y

0

Ψt(x)−p′ψt(x)

(

∫ y

x

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)p′

dx

)1p′(

∫ ∞

y

wt(x)dx

)1q

= supy∈(0,∞)

(

∫ y

0

V(x+ t)

(

∫ y

x

(

sups≤τU(τ+ t)

)

a(s+ t)ds

)p′

dx

)1p′(

∫ ∞

y

w(x+ t)dx

)1q

= supy∈(0,∞)

(

∫ y

0

V(x+ t)

(

∫ y

x

(

sups+t≤τU(τ)

)

a(s+ t)ds

)p′

dx

)1p′(

∫ ∞

y

w(x+ t)dx

)1q

16 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

= supy∈(0,∞)

(

∫ y

0

V(x+ t)

(

∫ y+t

x+t

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)1p′(

∫ ∞

y

w(x+ t)dx

)1q

= supy∈(0,∞)

(

∫ y+t

t

V(x)

(

∫ y+t

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)1p′(

∫ ∞

y+t

w(x)dx

)1q

= supy∈(t,∞)

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)1p′(

∫ ∞

y

w(x)dx

)1q

,

supy∈(0,∞)

(

∫ y

0

Ψt(x)−p′ψt(x)

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(z)dz

)1q

= supy∈(0,∞)

(

∫ y

0

V(x+ t)dx

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ+ t)

)

a(s+ t)ds

)q

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y

0

V(x+ t)dx

)1p′(

∫ ∞

y

(

∫ z

y

(

sups+t≤τU(τ)

)

a(s+ t)ds

)q

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y

0

V(x+ t)dx

)1p′(

∫ ∞

y

(

∫ z+t

y+t

(

sups≤τU(τ)

)

a(s)ds

)q

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y+t

t

V(x)dx

)1p′(

∫ ∞

y

(

∫ z+t

y+t

(

sups≤τU(τ)

)

a(s)ds

)q

w(z+ t)dz

)1q

= supy∈(0,∞)

(

∫ y+t

t

V(x)dx

)1p′(

∫ ∞

y+t

(

∫ z

y+t

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)1q

= supy∈(t,∞)

(

∫ y

t

V(x)dx

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)1q

,

supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)))1p′

(

∫ x

0

At(y)q wt(y)dy

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τU(τ+ t)p′

(

∫ τ

0

V(y+ t)dy

)))1p′

(

∫ x+t

t

(

∫ y+t

t

a

)q

w(y+ t)dy

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τU(τ+ t)p′

(

∫ τ+t

t

V(y)dy

)))1p′

(

∫ x+t

t

(

∫ y

t

a

)q

w(y)dy

)1q

= supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups+t≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ x+t

t

(

∫ y

t

a

)q

w(y)dy

)1q

= supx∈(0,∞)

(

∫

[x+t,∞)

d

(

− sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

)))1p′

(

∫ x+t

t

(

∫ y

t

a

)q

w(y)dy

)1q

= supx∈(t,∞)

(

∫

[x,∞)

d

(

− sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

)))1p′

(

∫ x

t

(

∫ y

t

a

)q

w(y)dy

)1q

,

supx∈(0,∞)

(

∫

(0,x]

At(s)p′ d

(

− sups≤τ

ut(τ)Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)))1p′

(

∫ ∞

x

wt(z)dz

)1q

= supx∈(0,∞)

(

∫

(0,x]

(

∫ s+t

t

a

)p′

d

(

− sups≤τ

U(τ+ t)p′(

∫ τ

0

V(y+ t)dy

)))1p′

(

∫ ∞

x

w(z+ t)dz

)1q

= supx∈(0,∞)

(

∫

(0,x]

(

∫ s+t

t

a

)p′

d

(

− sups≤τU(τ+ t)p′

(

∫ τ+t

t

V(y)dy

)))1p′

(

∫ ∞

x+t

w(z)dz

)1q

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 17

= supx∈(0,∞)

(

∫

(0,x]

(

∫ s+t

t

a

)p′

d

(

− sups+t≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ ∞

x+t

w(z)dz

)1q

= supx∈(0,∞)

(

∫

(t,x+t]

(

∫ s

t

a

)p′

d

(

− sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ ∞

x+t

w(z)dz

)1q

= supx∈(t,∞)

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

− sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

)))1p′

(

∫ ∞

x

w(z)dz

)1q

,

(

∫ ∞

0

At(z)q wt(z)dz

)1q

lims→∞

(

sups≤τ

ut(τ)Ψt(τ)2(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)1p′)

=

(

∫ ∞

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)1q

lims→∞

(

sups≤τU(τ+ t)

(

∫ τ

0

V(y+ t)dy

)1p′)

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τU(τ+ t)

(

∫ τ+t

t

V(y)dy

)1p′)

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups+t≤τU(τ)

(

∫ τ

t

V(y)dy

)1p′)

=

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τU(τ)

(

∫ τ

t

V(y)dy

)1p′)

,

(

∫ ∞

0

ψt(z)dz

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(x)dx

)1q

=

(

∫ ∞

0

V(z+ t)dz

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups≤τ

U(τ+ t)

)

a(s+ t)ds

)q

w(x+ t)dx

)1q

=

(

∫ ∞

t

V(z)dz

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups≤τ

U(τ+ t)

)

a(s+ t)ds

)q

w(x+ t)dx

)1q

=

(

∫ ∞

t

V(z)dz

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups+t≤τ

U(τ)

)

a(s+ t)ds

)q

w(x+ t)dx

)1q

=

(

∫ ∞

t

V(z)dz

)− 1p(

∫ ∞

0

(

∫ x+t

t

(

sups≤τ

U(τ)

)

a(s)ds

)q

w(x+ t)dx

)1q

=

(

∫ ∞

t

V(z)dz

)− 1p(

∫ ∞

t

(

∫ x

t

(

sups≤τ

W(τ)

)

a(s)ds

)q

w(x)dx

)1q

,

combining, we arrive at

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ y

t

(

sups≤τ u(τ)

(

∫ ∞

τf (z)dz

))

a(s)ds

)q

w(y)dy

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈ supy∈(t,∞)

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τ

U(τ)

)

a(s)ds

)p′

dx

)1p′(

∫ ∞

y

w(x)dx

)1q

+ supy∈(t,∞)

(

∫ y

t

V(x)dx

)1p′(

∫ ∞

y

(

∫ z

y

(

sups≤τ

U(τ)

)

a(s)ds

)q

w(z)dz

)1q

+ supx∈(t,∞)

(

∫

[x,∞)

d

(

− sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ x

t

(

∫ y

t

a

)q

w(y)dy

)1q

18 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+ supx∈(t,∞)

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

− sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

)))1p′

(

∫ ∞

x

w(z)dz

)1q

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τU(τ)

(

∫ τ

t

V(y)dy

)1p′)

+

(

∫ ∞

t

V(z)dz

)− 1p(

∫ ∞

t

(

∫ x

t

(

sups≤τU(τ)

)

a(s)ds

)q

w(x)dx

)1q

.

Let q < p. By [37, Theorem 3.3], we have for any t ∈ (0,∞) that

supf∈M+(0,∞)

(

∫ ∞

0

(

∫ y

0

(

sups≤τ ut(τ)

(

∫ ∞

τf (z)dz

))

at(s)ds

)q

wt(y)dy

)1q

(

∫ ∞

0f (s)pvt(s)ds

)1p

≈

(

∫ ∞

0

(

∫ y

0

Ψt(x)−p′ψt(x)

)

p(q−1)p−q

Ψt(y)−p′ψt(y)

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(z)dz

)

p

p−q

dy

)

p−q

pq

+

(

∫ ∞

0

(

∫ y

0

Ψt(x)−p′ψt(x)

(

∫ y

x

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

wt(z)dz

)

q

p−q

wt(y)dy

)

p−q

pq

+

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

))))

q(p−1)p−q

(

∫ x

0

At(z)q wt(z)dz

)

q

p−q

At(x)q wt(x)dx

)

p−q

pq

+

(

∫ ∞

0

(

∫

(0,x]

At(s)p′ d

(

−

(

sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

))))

q(p−1)p−q

(

∫ ∞

x

wt(z)dz

)

q

p−q

wt(x)dx

)

p−q

pq

+

(

∫ ∞

0

At(z)q wt(z)dz

)1q

lims→∞

(

sups≤τ

ut(τ)Ψt(τ)2(

∫ τ

0

Ψt(y)−p′ψt(y)dy

)1p′)

+

(

∫ ∞

0

ψt(t)dt

)− 1p(

∫ ∞

0

(

∫ x

0

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(x)dx

)1q

.

Since

(

∫ ∞

0

(

∫ y

0

Ψt(x)−p′ψt(x)

)

p(q−1)p−q

Ψt(y)−p′ψt(y)

(

∫ ∞

y

(

∫ z

y

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)q

wt(z)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

V(x+ t)dx

)

p(q−1)p−q

V(y+ t)

(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ+ t)

)

a(s+ t)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

V(x)dx

)

p(q−1)p−q

V(y+ t)

(

∫ ∞

y

(

∫ z

y

(

sups+t≤τU(τ)

)

a(s+ t)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

V(x)dx

)

p(q−1)p−q

V(y+ t)

(

∫ ∞

y

(

∫ z+t

y+t

(

sups≤τ

U(τ)

)

a(s)ds

)q

w(z+ t)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

V(x)dx

)

p(q−1)p−q

V(y+ t)

(

∫ ∞

y+t

(

∫ z

y+t

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

=

(

∫ ∞

t

(

∫ y

t

V(x)dx

)

p(q−1)p−q

V(y)

(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

,

(

∫ ∞

0

(

∫ y

0

Ψt(x)−p′ψt(x)

(

∫ y

x

(

sups≤τ

ut(τ)Ψt(τ)2)

at(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

wt(z)dz

)

q

p−q

wt(y)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

V(x+ t)

(

∫ y

x

(

sups≤τU(τ+ t)

)

a(s+ t)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z+ t)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 19

=

(

∫ ∞

0

(

∫ y

0

V(x+ t)

(

∫ y

x

(

sups+t≤τU(τ)

)

a(s+ t)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z+ t)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y

0

V(x+ t)

(

∫ y+t

x+t

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y+t

w(z)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

0

(

∫ y+t

t

V(x)

(

∫ y+t

x

(

sups≤τ

U(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y+t

w(z)dz

)

q

p−q

w(y+ t)dy

)

p−q

pq

=

(

∫ ∞

t

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

,

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

))))

q(p−1)p−q

(

∫ x

0

At(z)q wt(z)dz

)

q

p−q

At(x)q wt(x)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

sups≤τU(τ+ t)p′

(

∫ τ

0

V(y+ t)dy

))))

q(p−1)p−q

(

∫ x

0

(

∫ z+t

t

a

)q

w(z+ t)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

sups≤τU(τ+ t)p′

(

∫ τ+t

t

V(y)dy

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

sups+t≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

[x+t,∞)

d

(

−

(

sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ x+t

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x+t

t

a

)q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

sups≤τ

U(τ)p′(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

,

(

∫ ∞

0

(

∫

(0,x]

At(s)p′ d

(

−

(

sups≤τ

ut(τ)p′Ψt(τ)2p′(

∫ τ

0

Ψt(y)−p′ψt(y)dy

))))

q(p−1)p−q

(

∫ ∞

x

wt(z)dz

)

q

p−q

wt(x)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(0,x]

(

∫ s+t

t

a

)p′

d

(

−

(

sups≤τ

U(τ+ t)p′(

∫ τ

0

V(y+ t)dy

))))

q(p−1)p−q

(

∫ ∞

x

w(z+ t)dz

)

q

p−q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(0,x]

(

∫ s+t

t

a

)p′

d

(

−

(

sups+t≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ ∞

x+t

w(z)dz

)

q

p−q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

0

(

∫

(t,x+t]

(

∫ s

t

a

)p′

d

(

−

(

sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ ∞

x+t

w(z)dz

)

q

p−q

w(x+ t)dx

)

p−q

pq

=

(

∫ ∞

t

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

−

(

sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

qp−q

w(x)dx

)

p−qpq

,

combining, we arrive at

supf∈M+(t,∞)

(

∫ ∞

t

(

∫ y

t

(

sups≤τ u(τ)

(

∫ ∞

τf (z)dz

))

a(s)ds

)q

w(y)dy

)1q

(

∫ ∞

tf (s)pv(s)ds

)1p

≈

(

∫ ∞

t

(

∫ y

t

V(x)dx

)

p(q−1)p−q

V(y)

(

∫ ∞

y

(

∫ z

y

(

sups≤τU(τ)

)

a(s)ds

)q

w(z)dz

)

p

p−q

dy

)

p−q

pq

+

(

∫ ∞

t

(

∫ y

t

V(x)

(

∫ y

x

(

sups≤τU(τ)

)

a(s)ds

)p′

dx

)

q(p−1)p−q

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

20 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ x

t

(

∫ z

t

a

)q

w(z)dz

)

q

p−q(

∫ x

t

a

)q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫

(t,x]

(

∫ s

t

a

)p′

d

(

−

(

sups≤τU(τ)p′

(

∫ τ

t

V(y)dy

))))

q(p−1)p−q

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

t

(

∫ z

t

a

)q

w(z)dz

)1q

lims→∞

(

sups≤τU(τ)

(

∫ τ

t

V(y)dy

)1p′)

+

(

∫ ∞

t

v(s)1−p′ ds

)− 1p(p′+1)

(

∫ ∞

t

(

∫ x

t

(

sups≤τU(τ)

)

a(s)ds

)q

w(x)dx

)1q

.

The proof is completed.

3. Characterization of the restricted inequality

We start this section with some historical remarks concerning restricted inequalities related to the operator Tu,b.

Note that the inequality

(3.1) ‖Tu,b f ‖q,w,(0,∞) ≤C‖ f ‖p,v,(0,∞), f ∈M+,↓(0,∞)

was characterized in [24, Theorem 3.5] under condition

supt∈(0,∞)

u(t)

B(t)

∫ t

0

b(τ)

u(τ)dτ <∞.

However, the case when 0 < p ≤ 1 < q <∞ was not considered in [24]. It is also worth to mention that in the case

when 1 < p <∞, 0 < q < p<∞, q , 1 [24, Theorem 3.5] contains only discrete condition. In [25] the new reduction

theorem was obtained when 0 < p ≤ 1, and this technique allowed to characterize inequality (3.1) when b ≡ 1, and

in the case when 0 < q < p ≤ 1, [25] contains only discrete condition. The complete characterizations of inequality

(3.1) for 0 < q ≤ ∞, 0 < p ≤ ∞ were given in [18] and [36]. Using the results in [44–47], another characterization

of (3.1) was obtained in [55] and [49].

Now we present characterization of inequality (1.3).

Denote the best constant in inequality (1.3) by K, that is,

K := supf∈M+

(

∫ ∞

0

(

∫ x

0

[

Tu,b f ∗(t)]r

dt

)

q

r

w(x)dx

)1q

(

∫ ∞

0

(

∫ x

0[ f ∗(τ)]p dτ

)mp

v(x)dx

)1m

≡ supf∈M+

(

∫ ∞

0

(

∫ x

0

[

Tu,b f ∗(t)]r

dt

)

q

r

w(x)dx

)1q

‖ f ‖GΓ(p,m,v)

.

The following two reduction lemmas hold true.

Lemma 3.1. Let 1 < r < q <∞, 0 < p <∞, 0 < m <∞ and b ∈W(0,∞) be such that the function B(t) satisfies

0 < B(t) <∞ for every t ∈ (0,∞). Assume that u ∈W(0,∞)∩C(0,∞) and v, w ∈W(0,∞). Then

K = supg∈M+

1

‖g‖1r

qq−r ,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

supf∈M+

∫ ∞

0f ∗(y)b(y)

∫ ∞

yϕ(x)

u(x)B(x)

dxdy

‖ f ‖GΓ(p,m,v)

.

Proof. By duality, on using Fubini’s Theorem, we have that

K = supf∈M+

1

‖ f ‖GΓ(p,m,v)

supg∈M+

∫ ∞

0

(

∫ x

0

[

Tu,b f ∗(t)]r

dt

)

g(x)dx

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

1r

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 21

= supf∈M+

1

‖ f ‖GΓ(p,m,v)

supg∈M+

∫ ∞

0

(

∫ ∞

tg(x)dx

)[

supτ≥t

(

u(τ)B(τ)

∫ τ

0f ∗(y)b(y)dy

)r]

dt

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

1r

.

On using Theorem 2.3, we arrive at

K = supf∈M+

1

‖ f ‖GΓ(p,m,v)

supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

0

h(x)

(

u(x)

B(x)

∫ x

0

f ∗(y)b(y)dy

)r

dx

1r

.

By duality, we obtain that

K = supf∈M+

1

‖ f ‖GΓ(p,m,v)

supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

∫ ∞

0ϕ(x)

u(x)B(x)

∫ x

0f ∗(y)b(y)dydx

‖ϕ‖r′,h1−r′ ,(0,∞)

.

By Fubini’s Theorem, interchanging the suprema, we arrive at

K = supf∈M+

1

‖ f ‖GΓ(p,m,v)

supg∈M+

1

‖g‖1r

qq−r ,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

∫ ∞

0f ∗(y)b(y)

∫ ∞

yϕ(x)

u(x)B(x)

dxdy

‖ϕ‖r′,h1−r′ ,(0,∞)

= supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

supf∈M+

∫ ∞

0f ∗(y)b(y)

∫ ∞

yϕ(x)

u(x)B(x)

dxdy

‖ f ‖GΓ(p,m,v)

.

This completes the proof.

Lemma 3.2. Let 1< r < q<∞, 1< p<∞, 1<m<∞ and b ∈W(0,∞)∩M+((0,∞);↓) be such that the function B(t)

satisfies 0 < B(t) <∞ for every t ∈ (0,∞). Suppose that u ∈W(0,∞)∩C(0,∞), v ∈Wm,p(0,∞) and w ∈W(0,∞).

Denote by

(3.2) v2(t) :=t

m′

p′ v0(t)

v1(t)m′+1, t ∈ (0,∞),

where v0 and v1 are defined by (2.2) and (2.3), respectively. Then

K ≈ A+B,

where

A := supg∈M+

1

‖g‖1r

qq−r ,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,(

Bu

)r′

h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

ϕ

)p′(B(s)

s

)p′

ds

)m′

p′

v2(t)dt

)1

m′

and

B := supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,u−r′h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

ϕ

)p′ ds

sp′

)m′

p′

v2(t)dt

)1

m′

.

Proof. By Lemma 3.1 and Theorem 2.1, we have that

K ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

b(y)

(

∫ ∞

y

ϕu

B

)

dy

)p′

ds

)m′

p′

v2(t)dt

)1

m′

.

Since

1

s

∫ s

0

b(y)

(

∫ ∞

y

ϕ(x)u(x)

B(x)dx

)

dy =1

s

∫ s

0

b(y)

(

∫ s

y

ϕ(x)u(x)

B(x)dx

)

dy+1

s

∫ s

0

b(y)dy

∫ ∞

s

ϕ(x)u(x)

B(x)dx

22 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

=1

s

∫ s

0

ϕ(x)u(x)dx+B(s)

s

∫ ∞

s

ϕ(x)u(x)

B(x)dx,

we arrive at

K ≈ supg∈M+

1

‖g‖1r

q

q−r ,wr

r−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

( ∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

B(s)

s

∫ ∞

s

ϕ(x)u(x)

B(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

+ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

ϕ(x)u(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

.

Observe that the inequality

(

∫ ∞

0

(

∫ ∞

t

(

B(s)

s

∫ ∞

s

ϕ(x)u(x)

B(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≤ c

(

∫ ∞

0

ϕ(x)r′h(x)1−r′ dx

)1r′

holds true for all h ∈M+ if and only if the inequality

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

ϕ(x)dx

)p′(B(s)

s

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≤ c

(

∫ ∞

0

ϕ(x)r′(

B(x)

u(x)

)r′

h(x)1−r′ dx

)1r′

holds for all h ∈M+.

Similarly, the inequality

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

ϕ(x)u(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≤ c

(

∫ ∞

0

ϕ(x)r′h(x)1−r′ dx

)1r′

holds true for all h ∈M+ if and only if the inequality

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

ϕ(x)dx

)p′ ds

sp′

)m′

p′

v2(t)dt

)1

m′

≤ c

(

∫ ∞

0

ϕ(x)r′(

1

u(x)

)r′

h(x)1−r′ dx

)1r′

holds for all h ∈M+.

Thus,

K ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,(

Bu

)r′

h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

ϕ(x)dx

)p′(B(s)

s

)p′

ds

)m′

p′

v2(t)dt

)1

m′

+ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supϕ∈M+

1

‖ϕ‖r′,u−r′h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

ϕ(x)dx

)p′ ds

sp′

)m′

p′

v2(t)dt

)1

m′

= A+B

holds.

Theorem 3.3. Let 1 < m < p ≤ r < q <∞ and b ∈W(0,∞)∩M+((0,∞);↓) be such that the function B(t) satisfies

0 < B(t) <∞ for every t ∈ (0,∞). Suppose that u ∈W(0,∞)∩C(0,∞), v ∈Wm.p(0,∞) and w ∈W(0,∞). Suppose

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 23

that

0 <

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds <∞, t ∈ (0,∞),

where v2 is defined by (3.2). Then

K ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

∫ y

0

(

supx≤τ≤t

u(τ)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

supx≤τ≤t

u(τ)r)

dx

)1r(

∫ ∞

t

w(y)dy

)1q

.

Proof. By Lemma 3.2 we have that

K ≈ A+B.

We estimate A. By Theorem 2.7, (a), we have that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

ϕ(x)dx

)p′(B(s)

s

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

(

∫ ∞

t

h(x)

(

u(x)

B(x)

)r

dx

)1r(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

.

Hence

A ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supt∈(0,∞)

(

∫ ∞

t

h(x)

(

u(x)

B(x)

)r

dx

)1r(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

.

Interchanging the suprema, we get that

A ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

t

h(x)

(

u(x)

B(x)

)r

dx

1r

≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

0

h(x)

(

u(x)

B(x)

)r

χ(t,∞)(x)dx

1r

.

By Theorem 2.3, we have that

A ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

0

(

∫ ∞

x

g

)(

supx≤τ

(

u(τ)

B(τ)

)r

χ(t,∞)(τ)

)

dx

)1r

.

Applying Fubini’s Theorem, interchanging the suprema, by duality, we get that

A ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

supg∈M+

∫ ∞

0g(y)

∫ y

0

(

supx≤τ

(

u(τ)B(τ)

)r

χ(t,∞)(τ)

)

dxdy

‖g‖ q

q−r ,wr

r−q ,(0,∞)

1r

= supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

0

(

∫ y

0

(

supx≤τ

(

u(τ)

B(τ)

)r

χ(t,∞)(τ)

)

dx

)

q

r

w(y)dy

)1q

.

24 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

Since

supx≤τ

χ(t,∞)(τ)

(

u(τ)

B(τ)

)r

= supt≤τ

(

u(τ)

B(τ)

)r

,

when 0 < x ≤ t, we get that

A ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ t

0

(

∫ y

0

(

supx≤τ

(

u(τ)

B(τ)

)r

χ(t,∞)(τ)

)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

0

(

supx≤τ

(

u(τ)

B(τ)

)r

χ(t,∞)(τ)

)

dx

)

q

r

w(y)dy

)1q

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ t

0

yq

r w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′

t1r

(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

t

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

.

To estimate B, by Theorem 2.6, (a), we have that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

ϕ(x)dx

)p′ ds

sp′

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

(

∫ t

0

v2(τ)dτ

)1

m′(

∫ ∞

t

ds

sp′

)1p′(

∫ t

0

h(x)u(x)r dx

)1r

+ supt∈(0,∞)

(

∫ ∞

t

(

∫ ∞

s

dy

yp′

)m′

p′

v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

≈ supt∈(0,∞)

t− 1

p

(

∫ t

0

v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

+ supt∈(0,∞)

(

∫ ∞

t

s−m′

p v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

≈ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

.

Hence, interchanging the suprema, we get that

B ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

( ∫ ∞

τg)

dτ

supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

= supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′

sup

h:∫ x

0h≤

∫ x

0

( ∫ ∞

τg)

dτ

∫ ∞

0

h(x)u(x)rχ(0,t)(x)dx

1r

.

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 25

By Theorem 2.3, we have that

B ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ ∞

0

(

∫ ∞

x

g

)(

supx≤τ

u(τ)rχ(0,t)(τ)

)

dx

)1r

.

By Fubini’s Theorem, interchanging the suprema, by duality we get that

B ≈ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′

supg∈M+

∫ ∞

0g(y)

∫ y

0

(

supx≤τ u(τ)rχ(0,t)(τ)

)

dxdy

‖g‖ q

q−r,w

rr−q ,(0,∞)

1r

≈ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ ∞

0

(

∫ y

0

(

supx≤τ

u(τ)rχ(0,t)(τ)

)

dx

)

q

r

w(y)dy

)1q

.

Since

supx≤τ

u(τ)rχ(0,t)(τ) = supx≤τ≤t

u(τ)r

for 0 < x ≤ t, we arrive at

B ≈ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

∫ y

0

(

supx≤τ≤t

u(τ)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

supx≤τ≤t

u(τ)r)

dx

)1r(

∫ ∞

t

w(y)dy

)1q

.

Combining the estimates for A and B, we obtain that

K ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

∫ y

0

(

supx≤τ≤t

u(τ)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ t

0

(

supx≤τ≤t

u(τ)r)

dx

)1r(

∫ ∞

t

w(y)dy

)1q

.

Theorem 3.4. Let 1<m ≤ r <minp,q<∞ and b ∈W(0,∞)∩M+((0,∞);↓) be such that the function B(t) satisfies

0 < B(t) <∞ for every t ∈ (0,∞). Assume that u ∈W(0,∞)∩C(0,∞), v ∈Wm.p(0,∞) and w ∈W(0,∞). Suppose

that

0 <

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds <∞, t ∈ (0,∞),

∫ t

0

sm′

p′ v0(s)

v1(s)m′+1ds <∞,

∫ ∞

t

sm′(

1− 2p

)

v0(s)

v1(s)m′+1ds <∞ t ∈ (0,∞),

and

∫ 1

0

sm′

(

1− 2p

)

v0(s)

v1(s)m′+1ds =

∫ ∞

1

sm′

p′ v0(s)

v1(s)m′+1ds =∞,

where v0, v1 and v2 are defined by (2.2), (2.3) and (3.2), respectively. Denote by

V2(t) :=

(

∫ t

0

sm′

p′ v0(s)

v1(s)m′+1ds

)1

m′

, V3(t) =

(

∫ ∞

0

(

t

y+ t

)m′

p

v2(y)dy

)1

m′

, 0 < t <∞,

26 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

B(t, s) =

(

∫ s

t

(

B(y)

y

)p′

dy

)

p(r−1)p−r

(

B(s)

s

)p′

, 0 < t < s <∞,

and

K(τ, t) = u(τ)(τ+ t)− 2

2p−r , 0 < τ, t <∞.

(i) If p ≤ q, then

K ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)

(

∫ x

y

sups≤τ

(

u(τ)

B(τ)

)r

ds

)

pp−r

dy

)

p−rpr

(

∫ ∞

x

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)dy

)

p−rpr

(

∫ ∞

x

(

∫ y

x

(

sups≤τ

u(τ)

B(τ)

)r

ds

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

[x,∞)

d

(

− supt≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ x

t

(y− t)q

r w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

(t,x]

(y− t)p

p−r d

(

− supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ ∞

x

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

u(τ)

B(τ)

(

∫ τ

t

B(t,y)dy

)

p−r

pr)

+ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ s

y

(

supx≤τK(τ, t)r

)

dx

)

p

p−r

dy

)

p−r

pr(

∫ ∞

s

w(z)dz

)1q

+ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−r

pr(

∫ ∞

s

(

∫ y

s

(

supx≤τ

K(τ, t)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τ

K(τ, t)pr

p−r

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)))

p−r

pr(

∫ x

0

yq

r w(y)dy

)1q

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

(0,x]

sp

p−r d

(

− sups≤τ

K(τ, t)pr

p−r

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)))

p−r

pr(

∫ ∞

x

w(y)dy

)1q

+

(

∫ ∞

0

yq

r w(y)dy

)1q

supt∈(0,∞)

V3(t) lims→∞

(

sups≤τK(τ, t)

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−rpr

)

+ supt∈(0,∞)

V3(t) tr

p(2p−r)

(

∫ ∞

0

(

∫ x

0

(

sups≤τK(τ, t)r

)

ds

)

q

r

w(x)dx

)1q

.

(ii) If q < p, then

K ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−rpr

(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 27

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)ds

)

p(q−r)r(p−q)

B(t, τ)

(

∫ ∞

τ

(

∫ z

τ

sups≤y

(

u(y)

B(y)

)r

ds

)

q

r

w(z)dz

)

p

p−q

dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)

(

∫ τ

s

supz≤y

(

u(y)

B(y)

)r

dz

)

p

p−r

ds

)

q(p−r)r(p−q)

(

∫ ∞

τ

w(x)dx

)

q

p−q

w(τ)dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ x

t

(z− t)q

r w(z)dz

)

q

p−q

(x− t)q

r w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

(t,x]

(y− t)p

p−r d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr)

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ x

0

(τ+ t)p(3r−2p)

(2p−r)(p−r) dτ

)

p(q−r)r(p−q)

(x+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ ∞

x

(

∫ z

x

(

supy≤τK(τ, t)r

)

dy

)

q

r

w(z)dz

)

p

p−q

dx

)

p−q

pq

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ y

0

(x+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ y

x

(

sups≤τK(τ, t)r

)

ds

)

p

p−r

dx

)

q(p−r)r(p−q)

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τK(τ, t)

pr

p−r

(

∫ τ

0

(z+ t)p(3r−2p)

(2p−r)(p−r) dz

))))

q(p−r)r(p−q)

(

∫ x

0

zq

r w(z)dz

)

q

p−q

xq

r w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

(0,x]

yp

p−r d

(

−

(

supy≤τK(τ, t)

pr

p−r

(

∫ τ

0

(z+ t)p(3r−2p)

(2p−r)(p−r) dz

))))

q(p−r)r(p−q)

(

∫ ∞

x

w

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

0

yq

r w(y)dy

)1q

supt∈(0,∞)

V3(t) lims→∞

(

sups≤τ

K(τ, t)

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−r

pr)

+ supt∈(0,∞)

V3(t) tr

p(2p−r)

(

∫ ∞

0

(

∫ x

0

(

sups≤τ

K(τ, t)r)

ds

)

q

r

w(x)dx

)1q

.

Proof. Recall that, by Lemma 3.2, we know that

K ≈ A+B.

We estimate A. By Theorem 2.7, (b), we have that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ ∞

s

ϕ(x)dx

)p′(B(s)

s

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

h(x)

(

u(x)

B(x)

)r

dx

)1r

+ supt∈(0,∞)

(

∫ t

0

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ s

t

(

B(y)

y

)p′

dy

)

p(r−1)p−r

(

B(s)

s

)p′ (∫ ∞

s

h(x)

(

u(x)

B(x)

)r

dx

)

p

p−r

ds

)

p−r

pr

:= I + II.

Consequently, we get that

A ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

sup

h:∫ x

0h≤

∫ x

0

( ∫ ∞

τg)

dτ

I+ II

:= A1+A2.

It has been already shown in the proof of Theorem 3.3 that

A1 ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

28 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

.

Interchanging the suprema, we get that

A2 = supg∈M+

1

‖g‖1r

q

q−r ,wr

r−q ,(0,∞)

supt∈(0,∞)

V2(t) sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

(

∫ ∞

t

B(t, s)

(

∫ ∞

s

h(x)

(

u(x)

B(x)

)r

dx

)

p

p−r

ds

)

p−r

pr

.

By duality (recall that p/(p− r) = (p/r)′), we have that

(

∫ ∞

t

B(t, s)

(

∫ ∞

s

h(x)

(

u(x)

B(x)

)r

dx

)

p

p−r

ds

)

p−r

pr

=

supψ∈M+(t,∞)

∫ ∞

t

(

∫ ∞

sh(x)

(

u(x)B(x)

)r

dx

)

ψ(s)B(t, s)ds

(

∫ ∞

tψ(s)

pr B(t, s)ds

)rp

1r

.

Thus

A2 = supg∈M+

1

‖g‖1r

q

q−r ,wr

r−q ,(0,∞)

supt∈(0,∞)

V2(t)

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supψ∈M+(t,∞)

∫ ∞

t

(

∫ ∞

sh(x)

(

u(x)B(x)

)r

dx

)

ψ(s)B(t, s)ds

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

1r

.

By Fubini’s Theorem, interchanging the suprema, we get that

A2 = supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

(

∫ ∞

t

ψ(s)p

r B(t, s)ds

)− rp

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

t

h(x)

(

u(x)

B(x)

)r∫ x

t

ψ(s)B(t, s)dsdx

1r

.

By Theorem 2.3, we have that

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

t

h(x)

(

u(x)

B(x)

)r∫ x

t

ψ(s)B(t, s)dsdx

= sup

h:∫ x

0h≤

∫ x

0

( ∫ ∞

τg)

dτ

∫ ∞

0

h(x)χ(t,∞)(x)

(

u(x)

B(x)

)r∫ x

t

ψ(s)B(t, s)dsdx

=

∫ ∞

0

(

∫ ∞

x

g(z)dz

)

supx≤τ

χ(t,∞)(τ)

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdx

=

∫ t

0

(

∫ ∞

x

g(z)dz

)

supx≤τ

χ(t,∞)(τ)

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdx

+

∫ ∞

t

(

∫ ∞

x

g(z)dz

)

supx≤τ

χ(t,∞)(τ)

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdx.

Since

supx≤τ

χ(t,∞)(τ)

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds = supt≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds,

when 0 < x ≤ t, by Fubini’s Theorem, we get that

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

t

h(x)

(

u(x)

B(x)

)r∫ x

t

ψ(s)B(t, s)dsdx

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 29

= supt≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds

∫ t

0

(

∫ ∞

x

g(z)dz

)

dx

+

∫ ∞

t

(

∫ ∞

x

g(z)dz

)

supx≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdx

= t supt≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds

∫ ∞

t

g(z)dz

+ supt≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds

∫ t

0

g(z)zdz

+

∫ ∞

t

g(z)

∫ z

t

supx≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdxdz

≈ supt≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)ds

∫ ∞

0

g(z)

(

zt

z+ t

)

dz

+

∫ ∞

t

g(z)

∫ z

t

supx≤τ

(

u(τ)

B(τ)

)r∫ τ

t

ψ(s)B(t, s)dsdxdz

Thus

A2 ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

supt≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)ds

∫ ∞

0g(z)

(

ztz+t

)

dz

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

1r

+ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q ,(0,∞)

supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

∫ ∞

tg(z)

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdxdz

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

1r

.

Interchanging the suprema, we get that

A2 ≈ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)

supψ∈M+(t,∞)

∫ τ

tψ(s)B(t, s)ds

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

supg∈M+

∫ ∞

0g(z)

(

ztz+t

)

dz

‖g‖ q

q−r,w

rr−q ,(0,∞)

1r

+ supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

1(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

supg∈M+

∫ ∞

tg(z)

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdxdz

‖g‖ q

q−r,w

rr−q ,(0,∞)

1r

≈ supt∈(0,∞)

V2(t)

supt≤τ

(

u(τ)

B(τ)

)r

supψ∈M+(t,∞)

∫ ∞

tψ(s)B(t, s)χ(t,τ)(s)ds

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

supg∈M+

∫ ∞

0g(z)

(

ztz+t

)

dz

‖g‖ q

q−r,w

rr−q ,(0,∞)

1r

+ supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

1(

∫ ∞

tψ(s)

pr B(t, s)ds

)rp

supg∈M+

∫ ∞

0g(z)

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdxχ(t,∞)(z)dz

‖g‖ q

q−r,w

rr−q ,(0,∞)

1r

.

By duality, we arrive at

A2 ≈ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−rpr

(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

30 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+ supt∈(0,∞)

V2(t)

supψ∈M+(t,∞)

(

∫ ∞

t

(

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdx

)

q

r

w(z)dz

)rq

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

1r

.

(i) Let p ≤ q. By Theorem 2.8, we have for any t ∈ (0,∞) that

supψ∈M+(t,∞)

(

∫ ∞

t

(

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdx

)

q

r

w(z)dz

)rq

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

≈ supx∈(t,∞)

(

∫ x

t

B(t,y)

(

∫ x

y

sups≤τ

(

u(τ)

B(τ)

)r

ds

)

p

p−r

dy

)

p−r

p(

∫ ∞

x

w(y)dy

)rq

+ supx∈(t,∞)

(

∫ x

t

B(t,y)dy

)

p−rp

(

∫ ∞

x

(

∫ y

x

(

sups≤τ

u(τ)

B(τ)

)r

ds

)

q

r

w(y)dy

)rq

+ supx∈(t,∞)

(

∫

[x,∞)

d

(

− supt≤τ

(

u(τ)

B(τ)

)

prp−r

(

∫ τ

t

B(t,y)dy

)))

p−rp

(

∫ x

t

(y− t)q

r w(y)dy

)rq

+ supx∈(t,∞)

(

∫

(t,x]

(y− t)p

p−r d

(

− supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

p(

∫ ∞

x

w(y)dy

)rq

+

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

(

u(τ)

B(τ)

)r(∫ τ

t

B(t,y)dy

)

p−r

p)

.

Consequently,

A2 ≈ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)

(

∫ x

y

sups≤τ

(

u(τ)

B(τ)

)r

ds

)

p

p−r

dy

)

p−r

pr(

∫ ∞

x

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)dy

)

p−r

pr(

∫ ∞

x

(

∫ y

x

(

sups≤τ

u(τ)

B(τ)

)r

ds

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

[x,∞)

d

(

− supt≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ x

t

(y− t)q

r w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

(t,x]

(y− t)p

p−r d

(

− supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ ∞

x

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

u(τ)

B(τ)

(

∫ τ

t

B(t,y)dy

)

p−rpr

)

.

Combining, we arrive at

A ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)

(

∫ x

y

sups≤τ

(

u(τ)

B(τ)

)r

ds

)

pp−r

dy

)

p−rpr

(

∫ ∞

x

w(y)dy

)1q

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 31

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)dy

)

p−r

pr(

∫ ∞

x

(

∫ y

x

(

sups≤τ

u(τ)

B(τ)

)r

ds

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

[x,∞)

d

(

− supt≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ x

t

(y− t)qr w(y)dy

)1q

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

(t,x]

(y− t)p

p−r d

(

− supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t,y)dy

)))

p−r

pr(

∫ ∞

x

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

u(τ)

B(τ)

(

∫ τ

t

B(t,y)dy

)

p−r

pr)

.

(ii) Let q < p. By Theorem 2.8, we get for any t ∈ (0,∞) that

supψ∈M+(t,∞)

(

∫ ∞

t

(

∫ z

tsupx≤τ

(

u(τ)B(τ)

)r∫ τ

tψ(s)B(t, s)dsdx

)

q

r

w(z)dz

)rq

(

∫ ∞

tψ(s)

p

r B(t, s)ds

)rp

≈

(

∫ ∞

t

(

∫ τ

t

B(t, s)ds

)

p(q−r)r(p−q)

B(t, τ)

(

∫ ∞

τ

(

∫ z

τ

sups≤y

(

u(y)

B(y)

)r

ds

)

q

r

w(z)dz

)

p

p−q

dτ

)

r(p−q)pq

+

(

∫ ∞

t

(

∫ τ

t

B(t, s)

(

∫ τ

s

supz≤y

(

u(y)

B(y)

)r

dz

)

p

p−r

ds

)

q(p−r)r(p−q)

(

∫ ∞

τ

w(x)dx

)

q

p−q

w(τ)dτ

)

r(p−q)pq

+

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

prp−r

(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ x

t

(z− t)q

r w(z)dz

)

q

p−q

(x− t)q

r w(x)dx

)

r(p−q)pq

+

(

∫ ∞

t

(

∫

(t,x]

(y− t)p

p−r d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

r(p−q)pq

+

(

∫ ∞

t

(y− t)q

r w(y)dy

)rq

limx→∞

(

supx≤τ

(

u(τ)

B(τ)

)r(∫ τ

t

B(t, s)ds

)

p−r

p)

.

Thus

A2 ≈ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)ds

)

p(q−r)r(p−q)

B(t, τ)

(

∫ ∞

τ

(

∫ z

τ

sups≤y

(

u(y)

B(y)

)r

ds

)

q

r

w(z)dz

)

p

p−q

dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)

(

∫ τ

s

supz≤y

(

u(y)

B(y)

)r

dz

)

p

p−r

ds

)

q(p−r)r(p−q)

(

∫ ∞

τ

w(x)dx

)

q

p−q

w(τ)dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ x

t

(z− t)q

r w(z)dz

)

q

p−q

(x− t)q

r w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

(t,x]

(y− t)p

p−r d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr)

.

Combining, we arrive at

A ≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

supt≤τ

u(τ)

B(τ)

)(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

32 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)p′

dy

)m′

p′

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

u(τ)

B(τ)

)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)supt≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−r

pr(

∫ ∞

0

(

yt

y+ t

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)ds

)

p(q−r)r(p−q)

B(t, τ)

(

∫ ∞

τ

(

∫ z

τ

sups≤y

(

u(y)

B(y)

)r

ds

)

q

r

w(z)dz

)

p

p−q

dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)

(

∫ τ

s

supz≤y

(

u(y)

B(y)

)r

dz

)

p

p−r

ds

)

q(p−r)r(p−q)

(

∫ ∞

τ

w(x)dx

)

q

p−q

w(τ)dτ

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ x

t

(z− t)q

r w(z)dz

)

q

p−q

(x− t)q

r w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

(t,x]

(y− t)p

p−r d

(

−

(

supy≤τ

(

u(τ)

B(τ)

)

pr

p−r(

∫ τ

t

B(t, s)ds

))))

q(p−r)r(p−q)

(

∫ ∞

x

w(z)dz

)

q

p−q

w(x)dx

)

p−q

pq

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)q

r w(y)dy

)1q

limx→∞

(

supx≤τ

(

u(τ)

B(τ)

)(

∫ τ

t

B(t, s)ds

)

p−rpr

)

.

Now we estimate B. By Theorem 2.6, (b), and integrating by parts, we have that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

∫ s

0

ϕ(x)dx

)p′ ds

sp′

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

(

∫ t

0

v2(s)ds

)1

m′(

∫ ∞

t

(

∫ ∞

s

dy

yp′dy

)

r(p−1)p−r

(

∫ s

0

h(x)u(x)r dx

)r

p−r

h(s)u(s)r ds

)

p−r

pr

+ supt∈(0,∞)

(

∫ ∞

t

(

∫ ∞

s

dy

yp′

)m′

p′

v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

≈ supt∈(0,∞)

(

∫ t

0

v2(s)ds

)1

m′(

∫ ∞

t

sr

r−p

(

∫ s

0

h(x)u(x)r dx

)r

p−r

h(s)u(s)r ds

)

p−r

pr

+ supt∈(0,∞)

(

∫ ∞

t

s−m′

p v2(s)ds

)1

m′(

∫ t

0

h(x)u(x)r dx

)1r

.

Since(

∫ t

0

hur)

p

p−r

=

∫ t

0

d

(

∫ s

0

hur)

p

p−r

≈

∫ t

0

(

∫ s

0

hur)

rp−r

h(s)u(s)r ds,

then we get that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

ϕ(x)u(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

(

∫ t

0

v2(s)ds

)1

m′(

∫ ∞

t

sr

r−p

(

∫ s

0

h(x)u(x)r dx

)r

p−r

h(s)u(s)r ds

)

p−r

pr

+ supt∈(0,∞)

(

∫ ∞

t

s−m′

p v2(s)ds

)1

m′(

∫ t

0

(

∫ s

0

h(x)u(x)r dx

)r

p−r

h(s)u(s)r ds

)

p−rpr

.

By Lemma 2.4, we arrive at

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

ϕ(x)u(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 33

≈ supt∈(0,∞)

∫ ∞

0

t1p

t1p + s

1p

m′

v2(s)ds

1m′

∫ ∞

0

s1p

t1p + s

1p

pr

p−r(

s1p)−

pr

p−r

(∫ s

0

hur

)r

p−r

h(s)u(s)r ds

p−r

pr

≈ supt∈(0,∞)

(

∫ ∞

0

(

t

s+ t

)m′

p

v2(s)ds

)1

m′(

∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds

)

p−r

pr

.

Applying Theorem 2.5, it can be seen that if for some t > 0∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds <∞,

then∫ ∞

0

(

∫ s

0

hur)

pp−r

d

(

−

(

1

s+ t

)r

p−r)

≤p

p− r

∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds.

Similarly, if∫ ∞

0

(

∫ s

0

hur)

p

p−r

d

(

−

(

1

s+ t

)r

p−r)

<∞

for some t > 0, then∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds ≤p− r

p

∫ ∞

0

(

∫ s

0

hur)

p

p−r

d

(

−

(

1

s+ t

)r

p−r)

.

So, we can write∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds ≈

∫ ∞

0

(

∫ s

0

hur)

p

p−r

d

(

−

(

1

s+ t

)r

p−r)

.

We obtain that∫ ∞

0

(

1

s+ t

)r

p−r(

∫ s

0

hur)

rp−r

h(s)u(s)r ds ≈ −

∫ ∞

0

(

∫ s

0

hur)

p

p−r(

1

s+ t

)

2r−p

p−r(

−1

(s+ t)2

)

ds

=

∫ ∞

0

(

∫ s

0

hur)

p

p−r(

1

s+ t

)

p

p−r

ds.

Thus, we have that

supϕ∈M+

1

‖ϕ‖r′,h1−r′ ,(0,∞)

(

∫ ∞

0

(

∫ ∞

t

(

1

s

∫ s

0

ϕ(x)u(x)dx

)p′

ds

)m′

p′

v2(t)dt

)1

m′

≈ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ s

0

hur)

p

p−r(

1

s+ t

)

p

p−r

ds

)

p−r

pr

.

Consequenty, we get that

B ≈ supg∈M+

1

‖g‖1r

q

q−r ,wr

r−q

sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ s

0

hur)

p

p−r(

1

s+ t

)

p

p−r

ds

)

p−r

pr

.

Interchanging suprema, by duality (since p > r), we arrive at

B ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q

supt∈(0,∞)

V3(t) sup

h:∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

(

∫ ∞

0

(

∫ s

0

hur)

p

p−r(

1

s+ t

)

p

p−r

ds

)

p−r

pr

= supg∈M+

1

‖g‖1r

q

q−r,w

rr−q

supt∈(0,∞)

V3(t)

supψ∈M+(0,∞)

suph:

∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

0

(

∫ s

0hur

)

ψ(s)ds

(

∫ ∞

0ψ(s)

p

r (s+ t)p

r ds

)rp

1r

.

34 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

Applying Fubini Theorem, by Theorem 2.3, we have that

B ≈ supg∈M+

1

‖g‖1r

q

q−r,w

rr−q

supt∈(0,∞)

V3(t)

supψ∈M+(0,∞)

suph:

∫ x

0h≤

∫ x

0

(∫ ∞

τg)

dτ

∫ ∞

0h(x)u(x)r

(

∫ ∞

xψ

)

dx

(

∫ ∞

0ψ(s)

p

r (s+ t)p

r ds

)rp

1r

= supg∈M+

1

‖g‖1r

q

q−r,w

rr−q

supt∈(0,∞)

V3(t)

supψ∈M+(0,∞)

∫ ∞

0

(

∫ ∞

xg

)

supx≤τ u(τ)r(

∫ ∞

τψ

)

dx

(

∫ ∞

0ψ(s)

p

r (s+ t)p

r ds

)rp

1r

.

Again, by Fubini Theorem, and interchanging suprema, we have that

B ≈ supt∈(0,∞)

V3(t)

supψ∈M+(0,∞)

1(

∫ ∞

0ψ(s)

p

r (s+ t)p

r ds

)rp

supg∈M+

∫ ∞

0g(y)

∫ y

0supx≤τ u(τ)r

(

∫ ∞

τψ

)

dxdy

‖g‖ q

q−r,w

rr−q

1r

.

Applying duality principle yields the following estimate:

B ≈ supt∈(0,∞)

V3(t)

supψ∈M+(0,∞)

(

∫ ∞

0

(

∫ y

0supx≤τ u(τ)r

(

∫ ∞

τψ

)

dx

)

q

r

w(y)dy

)rq

(

∫ ∞

0ψ(s)

p

r (s+ t)p

r ds

)rp

1r

.

Here we apply Theorem 2.9.

(i) Let p ≤ q. Then we have

B ≈ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ s

y

(

supx≤τK(τ, t)r

)

dx

)

p

p−r

dy

)

p−r

pr(

∫ ∞

s

w(z)dz

)1q

+ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−r

pr(

∫ ∞

s

(

∫ y

s

(

supx≤τ

K(τ, t)r)

dx

)

q

r

w(y)dy

)1q

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τ

K(τ, t)pr

p−r

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)))

p−r

pr(

∫ x

0

yq

r w(y)dy

)1q

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

(0,x]

sp

p−r d

(

− sups≤τK(τ, t)

pr

p−r

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)))

p−rpr

(

∫ ∞

x

w(y)dy

)1q

+

(

∫ ∞

0

yq

r w(y)dy

)1q

supt∈(0,∞)

V3(t) lims→∞

(

sups≤τK(τ, t)

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−rpr

)

+ supt∈(0,∞)

V3(t) tr

p(2p−r)

(

∫ ∞

0

(

∫ x

0

(

sups≤τK(τ, t)r

)

ds

)

q

r

w(x)dx

)1q

.

(ii) Let q < p. Then we have

B ≈ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ x

0

(τ+ t)p(3r−2p)

(2p−r)(p−r) dτ

)

p(q−r)r(p−q)

(x+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ ∞

x

(

∫ z

x

(

supy≤τ

K(τ, t)r)

dy

)

q

r

w(z)dz

)

pp−q

dx

)

p−q

pq

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ y

0

(x+ t)p(3r−2p)

(2p−r)(p−r)

(

∫ y

x

(

sups≤τ

K(τ, t)r)

ds

)

pp−r

dx

)

q(p−r)r(p−q)

(

∫ ∞

y

w(z)dz

)

q

p−q

w(y)dy

)

p−q

pq

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τK(τ, t)

pr

p−r

(

∫ τ

0

(z+ t)p(3r−2p)

(2p−r)(p−r) dz

))))

q(p−r)r(p−q)

(

∫ x

0

zq

r w(z)dz

)

q

p−q

xq

r w(x)dx

)

p−q

pq

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 35

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

(0,x]

yp

p−r d

(

−

(

supy≤τK(τ, t)

pr

p−r

(

∫ τ

0

(z+ t)p(3r−2p)

(2p−r)(p−r) dz

))))

q(p−r)r(p−q)

(

∫ ∞

x

w

)

q

p−q

w(x)dx

)

p−q

pq

+

(

∫ ∞

0

yq

r w(y)dy

)1q

supt∈(0,∞)

V3(t) lims→∞

(

sups≤τK(τ, t)

(

∫ τ

0

(y+ t)p(3r−2p)

(2p−r)(p−r) dy

)

p−r

pr)

+ supt∈(0,∞)

V3(t) tr

p(2p−r)

(

∫ ∞

0

(

∫ x

0

(

sups≤τK(τ, t)r

)

ds

)

q

r

w(x)dx

)1q

.

The proof is completed.

4. Boundedness of Mφ,Λα(b) from GΓ(p1,m1,v) into GΓ(p2,m2,w)

In this section we give the proof of the reduction theorem for the boundedness of Mφ,Λα(b) from GΓ(p1,m1,v1)

into GΓ(p2,m2,v2) and calculate the best constant in inequality (1.4).

Proof of Theorem 1.1:

Assume that the inequality

(

∫ ∞

0

(

∫ x

0

[(

Mφ,Λα(b) f)∗

(t)]p2 dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p1 dτ

)

m1p1

v(x)dx

)1

m1

holds for all f ∈M(Rn).

Denote byMrad,↓(Rn) the set of all measurable, non-negative, radially decreasing functions on Rn, that is,

Mrad,↓(Rn) := f ∈M(Rn) : f (x) = h(|x|), x ∈ Rn with h ∈M+((0,∞);↓),

where the notation M+((0,∞);↓) is used to denote the subset of those functions from M+(0,∞) which are non-

increasing on (0,∞).

Recall that the inequality

(

Mφ,Λα(b)g)∗

(t) ≥C supt≤τ

φ(τ)−1(

∫ τ

0

[g∗(y)]αb(y)dy

)1α

holds for all g ∈Mrad,↓(Rn) with constant C > 0 independent of g and t (cf. [36, Lemma 3.12]).

Thus the inequality

(

∫ ∞

0

(

∫ x

0

[

supt≤τ

φ(τ)−1(

∫ τ

0

[g∗(y)]αb(y)dy

)1α]p2

dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[g∗(τ)]p1 dτ

)

m1p1

v(x)dx

)1

m1

holds for all g ∈Mrad,↓(Rn), which evidently can be rewritten as follows

(

∫ ∞

0

(

∫ x

0

[

(TB/φα,bg∗)(t)]

p2α dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[g∗(τ)]p1α dτ

)

m1p1

v(x)dx

)1

m1, g ∈Mrad,↓(Rn).

Since for any h ∈M(Rn) there exists g ∈Mrad,↓(Rn) such that g∗ = h∗, then the ineqaulity

(

∫ ∞

0

(

∫ x

0

[

(TB/φα,bh∗)(t)]

p2α dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[h∗(τ)]p1α dτ

)

m1p1

v(x)dx

)1

m1

holds for all h ∈M(Rn), as well.

Now assume that the inequality

(

∫ ∞

0

(

∫ x

0

[

(TB/φα,bh∗)(t)]

p2α dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[h∗(τ)]p1α dτ

)

m1p1

v(x)dx

)1

m1

holds for all h ∈M(Rn).

Obviously, the last inequality is equivalent to the inequality

(

∫ ∞

0

(

∫ x

0

[

supt≤τ

φ(τ)−1(

∫ τ

0

[ f ∗(y)]αb(y)dy

)1α]p2

dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p1 dτ

)

m1p1

v(x)dx

)1

m1

for all f ∈M(Rn).

36 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

Recall that the inequality

(Mφ,Λα(b) f )∗(t) ≤C supt≤τ

φ(τ)−1(

∫ τ

0

[ f ∗(y)]αb(y)dy

)1α

holds for all f ∈M(Rn) (cf. [36, Corollary 3.6]).

Consequently, the inequality

(

∫ ∞

0

(

∫ x

0

[(

Mφ,Λα(b) f)∗

(t)]p2 dt

)

m2p2

w(x)dx

)1

m2≤C

(

∫ ∞

0

(

∫ x

0

[ f ∗(τ)]p1 dτ

)

m1p1

v(x)dx

)1

m1

holds for all f ∈M(Rn), as well.

The proof is completed.

Denote by

(4.1) v0(t) := tm1p1−1

∫ t

0

v(s)sm1p1 ds

∫ ∞

t

v(s)ds, t ∈ (0,∞),

(4.2) v1(t) :=

∫ t

0

v(s)sm1p1 ds+ t

m1p1

∫ ∞

t

v(s)ds, t ∈ (0,∞),

and

(4.3) v2(t) :=t

m1(p1−α)

p1(m1−α) v0(t)

v1(t)2m1−α

m1−α

, t ∈ (0,∞).

Theorem 4.1. Let 0 < α < m1 < p1 ≤ p2 < m2 < ∞, α ≤ r < ∞ and b ∈ W(0,∞)∩M+((0,∞);↓) be such that

the function B(t) satisfies 0 < B(t) <∞ for every t ∈ (0,∞), B ∈ ∆2, B(∞) =∞ and B(t)/tα/r is quasi-increasing.

Moreover, let φ ∈ W(0,∞)∩C(0,∞) be such that φ ∈ Qr(0,∞) is a quasi-increasing function. Assume that v ∈

Wm1 ,p1(0,∞) and w ∈W(0,∞). Suppose that

0 <

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds <∞, t ∈ (0,∞),

where v2 is defined by (4.3). Then

‖Mφ,Λα(b)‖GΓ(p1,m1,v)→GΓ(p2,m2,w)

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

supt≤τ

1

φ(τ)

)

∫ ∞

0

(

yt

y+ t

)

m2p2

w(y)dy

1m2

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α

∫ ∞

t

(

∫ y

t

(

supx≤τ

(

1

φ(τ)p2

))

dx

)

m2p2

w(y)dy

1m2

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)

m1α

p1(m1−α)

v2(s)ds

)

m1−α

m1α

∫ t

0

(

∫ y

0

(

supx≤τ≤t

(

B(τ)

φ(τ)α

)

p2α)

dx

)

m2p2

w(y)dy

1m2

+ supt∈(0,∞)

(

∫ ∞

0

(

1

s+ t

)

m1α

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

∫ y

0

(

supx≤τ≤t

(

B(τ)

φ(τ)α

)

p2α)

dx

)1p2

(∫ ∞

t

w(y)dy

)1

m2

.

Proof. The statement immediately follows by Theorems 1.1 and 3.3.

Theorem 4.2. Let 0 < α < m1 ≤ p2 < minp1,m2 <∞, α ≤ r <∞ and b ∈W(0,∞)∩M+((0,∞);↓) be such that

the function B(t) satisfies 0 < B(t) <∞ for every t ∈ (0,∞), B ∈ ∆2, B(∞) =∞ and B(t)/tα/r is quasi-increasing.

Moreover, let φ ∈ W(0,∞)∩C(0,∞) be such that φ ∈ Qr(0,∞) is a quasi-increasing function. Assume that v ∈

Wm1 ,p1(0,∞) and w ∈W(0,∞). Suppose that

0 <

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds <∞, t ∈ (0,∞),

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 37

∫ t

0

sm1(p1−α)

p1(m1−α) v0(s)v1(s)−

2m1−α

m1−α ds <∞,

∫ ∞

t

sm1(p1−2α)

p1(m1−α) v0(s)v1(s)−

2m1−α

m1−α ds <∞, t ∈ (0,∞),

and

∫ 1

0

sm1(p1−2α)

p1(m1−α) v0(s)v1(s)−

2m1−α

m1−α ds =

∫ ∞

1

sm1(p1−α)

p1(m1−α) v0(s)v1(s)−

2m1−α

m1−α ds =∞,

where v0, v1 and v2 are defined by (4.1), (4.2) and (4.3), respectively. Denote by

V2(t) :=

(

∫ t

0

sm1(p1−α)

p1(m1−α) v0(s)v1(s)−

2m1−α

m1−α ds

)

m1−α

m1α

, V3(t) =

(

∫ ∞

0

(

t

y+ t

)

m1α

p1(m1−α)

v2(y)dy

)

m1−α

m1α

, 0 < t <∞,

B(t, s) =

(

∫ s

t

(

B(y)

y

)

p1p1−α

dy

)

p1(p2−α)

(p1−p2)α(

B(s)

s

)

p1p1−α

, 0 < t < s <∞,

and

K(τ, t) =B(τ)

1α

φ(τ)(τ+ t)

− 22p1−p2 , 0 < τ, t <∞.

(i) If p1 ≤ m2, then

‖Mφ,Λα(b)‖GΓ(p1,m1,v)→GΓ(p2,m2,w)

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

supt≤τ

1

φ(τ)

)(

∫ ∞

0

(

yt

y+ t

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

∫ ∞

t

(

∫ y

t

(

supx≤τ

1

φ(τ)

)p2

dx

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t) supt≤τ

1

φ(τ)

(

∫ τ

t

B(t, s)ds

)

p1−p2p1 p2

(

∫ ∞

0

(

yt

y+ t

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)

(

∫ x

y

(

sups≤τ

1

φ(τ)

)p2

ds

)

p1p1−p2

dy

)

p1−p2p1 p2

(

∫ ∞

x

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫ x

t

B(t,y)dy

)

p1−p2p1 p2

(

∫ ∞

x

(

∫ y

x

(

sups≤τ

1

φ(τ)

)p2

ds

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

[x,∞)

d

(

− supt≤τ

(

1

φ(τ)

)

p1 p2p1−p2

(

∫ τ

t

B(t,y)dy

)))

p1−p2p1 p2

(

∫ x

t

(y− t)m2p2 w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t) supx∈(t,∞)

(

∫

(t,x]

(y− t)p1

p1−p2 d

(

− supy≤τ

(

1

φ(τ)

)

p1 p2p1−p2

(

∫ τ

t

B(t,y)dy

)))

p1−p2p1 p2

(

∫ ∞

x

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)m2p2 w(y)dy

)1

m2limx→∞

(

supx≤τ

1

φ(τ)

(

∫ τ

t

B(t,y)dy

)

p1−p2p1 p2

)

+ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2)

(

∫ s

y

supx≤τK(τ, t)p2 dx

)

p1p1−p2

dy

)

p1−p2p1 p2

(

∫ ∞

s

w(z)dz

)1

m2

+ supt∈(0,∞)

V3(t) sups∈(0,∞)

(

∫ s

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dy

)

p1−p2p1 p2

(

∫ ∞

s

(

∫ y

s

supx≤τ

K (τ, t)p2 dx

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

[x,∞)

d

(

− sups≤τK(τ, t)

p1 p2p1−p2

(

∫ τ

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dy

)))

p1−p2p1 p2

(

∫ x

0

ym2p2 w(y)dy

)1

m2

+ supt∈(0,∞)

V3(t) supx∈(0,∞)

(

∫

(0,x]

sp1

p1−p2 d

(

− sups≤τK(τ, t)

p1 p2p1−p2

(

∫ τ

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dy

)))

p1−p2p1 p2

(

∫ ∞

x

w(y)dy

)1

m2

38 R.CH. MUSTAFAYEV, N. BILGICLI, AND M. YILMAZ

+

(

∫ ∞

0

ym2p2 w(y)dy

)1

m2sup

t∈(0,∞)

V3(t) lims→∞

(

sups≤τ

K(τ, t)

(

∫ τ

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dy

)

p1−p2p1 p2

)

+ supt∈(0,∞)

V3(t) tp2

p1(2p1−p2)

(

∫ ∞

0

(

∫ x

0

sups≤τK(τ, t)p2 ds

)

m2p2

w(x)dx

)1

m2.

(ii) If m2 < p1, then

‖Mφ,Λα(b)‖GΓ(p1,m1,v)→GΓ(p2,m2,w)

≈ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

supt≤τ

1

φ(τ)

)(

∫ ∞

0

(

yt

y+ t

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

(

∫ t

0

(

∫ t

s

(

B(y)

y

)

p1p1−α

dy

)

m1(p1−α)

p1(m1−α)

v2(s)ds

)

m1−α

m1α(

∫ ∞

t

(

∫ y

t

(

supx≤τ

1

φ(τ)

)p2

dx

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t)supt≤τ

(

1

φ(τ)

)(

∫ τ

t

B(t, s)ds

)

p1−p2p1 p2

(

∫ ∞

0

(

yt

y+ t

)

m2p2

w(y)dy

)1

m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)ds

)

p1(m2−p2)

p2(p1−m2)

B(t, τ)

(

∫ ∞

τ

(

∫ z

τ

(

supx≤y

1

φ(y)

)p2

dx

)

m2p2

w(z)dz

)

p1p1−m2

dτ

)

p1−m2p1m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫ τ

t

B(t, s)

(

∫ τ

s

(

supz≤y

1

φ(y)

)p2

dz

)

p1p1−p2

ds

)

m2(p1−p2)

p2(p1−m2)(

∫ ∞

τ

w(x)dx

)

m2p1−m2

w(τ)dτ

)

p1−m2p1m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

[x,∞)

d

(

−

(

supy≤τ

(

1

φ(τ)

)

p1 p2(p1−p2)

(

∫ τ

t

B(t, s)ds

))))

m2(p1−p2)

p2(p1−m2)

×

×

(

∫ x

t

(z− t)m2p2 w(z)dz

)

m2p1−m2

xm2p2 w(x)dx

)

p1−m2p1m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(

∫

(t,x]

(y− t)p1

p1−p2 d

(

−

(

supy≤τ

(

1

φ(τ)

)

p1 p2(p1−p2)

(

∫ τ

t

B(t, s)ds

))))

m2(p1−p2)

p2(p1−m2)

×

×

(

∫ ∞

x

w(z)dz

)

m2p1−m2

w(x)dx

)

p1−m2p1m2

+ supt∈(0,∞)

V2(t)

(

∫ ∞

t

(y− t)m2p2 w(y)dy

)1

m2limx→∞

(

supx≤τ

1

φ(τ)

(

∫ τ

t

B(t, s)ds

)

p1−p2p1 p2

)

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ x

0

(τ+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dτ

)

p1(m2−p2)

p2(p1−m2)

(x+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2)×

×

(

∫ ∞

x

(

∫ z

x

(

supy≤τ

K (τ, t)p2

)

dy

)

m2p2

w(z)dz

)

p1p1−m2

dx

)

p1−m2p1m2

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫ y

0

(x+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2)

(

∫ y

x

(

sups≤τ

K (τ, t)p2

)

ds

)

p1p1−p2

dx

)

m2(p1−p2)

p2(p1−m2)

×

×

(

∫ ∞

y

w(z)dz

)

m2p1−m2

w(y)dy

)

p1−m2p1m2

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

[x,∞)

d

(

−

(

supy≤τK(τ, t)

p1 p2p1−p2

(

∫ τ

0

(z+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dz

))))

m2(p1−p2)

p2(p1−m2)

×

×

(

∫ x

0

zm2p2 w(z)dz

)

m2p1−m2

xm2p2 w(x)dx

)

p1−m2p1m2

ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 39

+ supt∈(0,∞)

V3(t)

(

∫ ∞

0

(

∫

(0,x]

yp1

p1−p2 d

(

−

(

supy≤τ

K(τ, t)p1 p2

p1−p2

(

∫ τ

0

(z+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dz

))))

m2(p1−p2)

p2(p1−m2)

×

×

(

∫ ∞

x

w(z)dz

)

m2p1−m2

w(x)dx

)

p1−m2p1m2

+

(

∫ ∞

0

ym2p2 w(y)dy

)1

m2sup

t∈(0,∞)

V3(t) lims→∞

(

sups≤τK(τ, t)

(

∫ τ

0

(y+ t)p1(3p2−2p1)

(2p1−p2)(p1−p2) dy

)

p1−p2p1 p2

)

+ supt∈(0,∞)

V3(t)tp2

p1(2p1−p2)

(

∫ ∞

0

(

∫ x

0

(

sups≤τK(τ, t)p2

)

ds

)

m2p2

w(x)dx

)1

m2.

Proof. The statement immediately follows by Theorems 1.1 and 3.4.

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ON SOME RESTRICTED INEQUALITIES FOR Tu,b AND THEIR APPLICATIONS 41

RZA MUSTAFAYEV, Department of Mathematics, Kamil Ozdag Faculty of Science, Karamanoglu Mehmetbey University, 70200,

Karaman, Turkey

Email address: rzamustafayev@gmail.com

NEVIN BILGICLI, Republic of TurkeyMinistry of National Education, Kirikkale High School, 71100, Kirikkale, Turkey

Email address: nevinbilgicli@gmail.com

MERVE YILMAZ, Department ofMathematics, Kamil Ozdag Faculty of Science, KaramanogluMehmetbeyUniversity, 70200, Kara-

man, Turkey

Email address: mervegorgulu@kmu.edu.tr