embedding derivatives of weighted hardy spaces into lebesgue spaces

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Embedding derivatives of weighted Hardy spaces into Lebesgue spaces

Daniel Girela, María Lorente and María Dolores Sarrión

Mathematical Proceedings of the Cambridge Philosophical Society / Volume 116 / Issue 01 / July 1994, pp 151 166DOI: 10.1017/S0305004100072455, Published online: 24 October 2008

Link to this article: http://journals.cambridge.org/abstract_S0305004100072455

How to cite this article:Daniel Girela, María Lorente and María Dolores Sarrión (1994). Embedding derivatives of weighted Hardy spaces into Lebesgue spaces. Mathematical Proceedings of the Cambridge Philosophical Society, 116, pp 151166 doi:10.1017/S0305004100072455

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Math. Proc. Camb. Phil. Soc. (1994), 116, 151 1 5 1

Printed in Great Britain

Embedding derivatives of weighted Hardy spaces intoLebesgue spaces1

BY DANIEL GIRELA, MARIA LORENTE

Andlisis Matemdtico, Facultad de Ciencias, Universidad de Malaga,29071 Malaga, Spain

AND MARIA DOLORES SARRION

Departamento de Economia Aplicada (Estad. y Econom.), Facultad de C. Economicasy Empresariales, Universidad de Malaga, 29071 Malaga, Spain

{Received 1 May 1993; revised 9 September 1993)

Abstract

Let 1 ̂ p < oo and let it; be a non-negative function defined on the unit circle Twhich satisfies the Ap condition of Muckenhoupt. The weighted Hardy space Hp(iv)consists of those functions / in the classical Hardy space H1 whose boundary valuesbelong to Lp(w). Recently McPhail (Studia Math. 96, 1990) has characterized thosepositive Borel measures /i on the unit disc A for which Hp(w) is continuouslycontained in Lv(d[i). In this paper we study the question of finding necessary andsufficient conditions on a positive Borel measure fi on A for the differentiationoperator/) denned byDf = f to map/P(w;) continuously intoLp(d/i). We prove thata necessary condition is that there exists a positive constant C such that

/J,(S(I)) < C\I\P I w{ei6)dd, for every interval / c T, (A)J i

where for any interval / c T ,

{z = rei0:eiOel, 0 < 1-r < min(L |

We prove that this condition is also sufficient in some cases, namely for 2 < p <oo and iv(ew) = \d\a, (\6\ ^n), — 1 < a < p— 1, but not in general. In the general casewe prove the sufficiency of a condition which is slightly stronger than (A).

1. Introduction and main results

Let A denote the unit disc {zeC: \z\ < 1} and T the unit circle {£eC: |£| = 1}. For0 < p < oo the Hardy space Hp consists of those functions/ analytic in A for which

11/11//' = sup L - \f(reie)\pdd) <oo.0<r<l \^n J-n I

1 This research has been supported in part by a D.G.I.C.Y.T. grant (PB91-0413) and by grantsfrom 'La Junta de Andalucia'.

152 DANIEL GIRELA, MARIA LORENTE AND MARIA DOLORES SARRION

A positive Borel measure /i on A is said to be a Carleson measure if there exists apositive constant A such that for every interval / of T

lx(S(I)) ^ A\I\, (1-1)

where |/| denotes the length of / and

S(I)={z = reie:eieeI, 0 < 1-r < min(l,|/|)}. (1-2)

A well known result of Carleson[1, 2] (see also [3], theorem 93) asserts that if 0 <p < oo and ji is a positive Borel measure on A then JX is a Carleson measure if and only\£HP cz Lp(d/A,) and the injection mapping from Hv to the space Lp(dfi) is continuous.

The question of characterizing those positive Borel measures fi on A for which thedifferentiation operator D denned by Df = / ' maps Hp continuously into Lp(dfi) iscompletely solved. The case p ^ 2 is due to Shirokov[ll] and Luecking[8] whoproved the following.

THEOREM A. Let 2 ̂ p < oo and let /A be a positive Boreal measure on A. Then thedifferentiation operator D defined by Df = / ' maps Hp continuously into Lp(dju) if andonly if there exists a positive constant C such that

fi(S(I)) s$ C\I\P+1, for every interval I cz T. (1-3)

If 0 < p < 2 then condition (l-3) is necessary for D to map Hp continuously intoLp(d/i) but it is not sufficient as the functions constructed in [5] and [6] prove. Afterthe publication of [5] we became aware of the very complete work of Luecking [8]who, among many other results, gave a necessary and sufficient condition on ft forthe operator D to apply Hp continuously into Lp(d/i).

Now let 1 ̂ p < oo and let w be a non-negative function denned on T whichsatisfies the^4p condition of Muckenhoupt (or simply weAp) [10] (see also chapter IVof [4] or chapter IX of [12]), that is, there exists a positive constant C such that, forevery interval / cz T,

j f w{eie)dd\l~ f wie*8)-1""-"ddT ' ^ C, if 1 < p < oo, (1-4)

or 4r \ w(ew)dd ^Cessinfw(eie), if p = l. (1-5)\I\Jl elsel

We define Lp(w) as the space of all complex-valued Lebesgue measurable functions/ on T for which

1/p

<oo. (1-6)

It is easy to see that, since weAp, Lp(w) c i ' f T ) . On the other hand, for eachfunction/e//1, the non-tangential limit f(eie) exists almost everywhere and/(e'")eTv^T) [3, theorem 2-2]. Consequently, we may define the weighted Hardy spaceHp(w) as follows

Hp(w) = {feHl:f{eie) eLp(w)} (1-7)equipped with the norm

ll/li l l / ( w ) W (1-8)Notice that when w = 1, Hv(w) coincides with the classical Hardy space Hp. With adifferent notation, the spaces Lp(w) and Hv(w) were considered by McPhail in [9] who

Weighted Hardy spaces 153

extended Carleson's Theorem obtaining a characterization of those positive Borelmeasures /ion A for which Hp(io) c Lp(dfi) and the injection mapping from Hp(w)into Lp(dfi) is continuous. Namely, he proved the following results.

THEOREM B [9, theorems 3 and 4]. Let 1 ̂ p < oo, weAp and let /i be a positiveBorel measure on A. If 1 < p < oo, then the following three conditions are equivalentwhile if p = 1 then the conditions (i) and (iii) are equivalent.

(i) For every feLp(w), the Poisson integral P/ off belongs to Lp(dfi) and the operatorP maps Lp(w) continuously into Lp(d/i), i.e. there exists a positive constant G such that

Wf\\L*>W^C\\f\\Lv(w), for every feLp(w). (1-9)

(ii) Hp(w) a Lp(dfi) and the injection mapping from Hp(w) into Lp(d/i) is continuous,i.e. there exists a positive constant C such that

\\f\\L"W^C\\f\\H,(w), for every feHp(w). (MO)

(iii) There exists a positive constant C such that

fi{S(I)) ^ C w(eie)d6, for every interval I <= T. (1-11)

We remark that our statement of Theorem B is slightly different from that of [9].First of all, with respect to the notation, notice that what we call w is denoted by vp

in [9] and our ||. \\Lp(w) is the ||. \\LP{V) of [9], Also, in the case 1 < p < oo [9] only statesthe equivalence of (i) and (iii), however, it is easy to see that in this case (i) and (ii)are equivalent. Indeed, the implication (i) => (ii) is trivial and the implication (ii) =>(i) follows from a well known result of Hunt, Muckenhoupt and Wheeden [7, theorem1] which implies that, since weAp and 1 < p < oo, the conjugation operator isbounded from Lp(w) into itself. Also, we should mention that throughout the paperwe are using the convention that C denotes a numerical constant not necessarily thesame in each instance.

In view of these results, the main object of this paper is to study the followingquestion: given 1 ̂ p < oo and weAp, for what positive Borel measures fi on A is ittrue that the differentiation operator/-^/' maps Hp(w) continuously into Lp(dfi) 1 InTheorem 1 we give a necessary condition.

THEOREM 1. Let 1 < p < oo and weAp. Let /ibea positive Borel measure on A. If thedifferentiation operator D defined by Df = f maps Hp(w) continuously into Lp(dfi) thenthere exists a positive constant C such that

rC\I\P \ w(ew)d6, for every interval I c T. (M2)

Ji

Now we turn to look for conditions on /i which are sufficient for the differentiationoperator!) to map Hp(w) continuously into Lp(d/i). We shall prove in Theorem 2 andTheorem 3 that condition (1-12) is close to being sufficient. In order to state ourresults concisely, let us introduce the following terminology. A function p: (0,2n] ->[0, oo) is said to be a Dini function if

— eft < oo, for some ft > 0. (1-13)

154 D A N I E L G I R E L A , M A R I A L O R E N T E AND MARIA D O L O R E S SARRION

Typical examples of bounded Dini functions are the functions p(t) = tf (e > 0) andp(t) — min(l, |log£|~a) (a > 1). Now we can state the following result.

THEOREM 2. Let 1 <p < oo, weAp and let p: ( 0 ,2T7]^ [0 , oo) be a bounded Dinifunction. Let ju be a positive Borel measure on A and suppose that there exists a positiveconstant C such that

fi{S(I))^C\I\pp{\I\)\ w{eie)dd, for every interval I c T. (1-14)

Then the differentiation operator D maps Hp(w) continuously into Lp(d/i).

For p = 1 we have the following weaker result.

THEOREM 3. Let weA1 and let /ibea positive Borel measure on A. Suppose that thereexist e > 0 and a positive constant C such that

rfi(S(I)) ^ C\I\1+e w(ew) dd, for every interval I c T. (1-15)

J i

Then the differentiation operator D maps H^w) continuously into L1{d/i).

Now let us turn our attention to the following question: Let 1 ^ p < oo andweAp. Is condition (1*12) sufficient for the differentiation operator/) to map Hp(w)continuously into Lp(d/i) ? As we have already mentioned in the case w = 1 and 1 ^p < 2, the answer to this question is negative. However, Theorem A shows that theanswer is affirmative in the case w = 1 and 2 < p < oo. This leaves open the questionof whether or not (1-12) is sufficient for the differentiation operator/) to map Hp(w)continuously into Lp(d/i) for 2 ^ p < oo and general weAp. We can give only apartial answer to this question. It is well known that if 1 < p < oo and wa(e

ie) = |#|a

(\6\ < n) then waeAp if and only if — 1 < a <p— 1 (see e.g. [12, p. 236]). We canprove the following result which generalizes Theorem A.

THEOREM 4. Let 2 ^ p < oo, — i <<z <<p— 1 and wa{eie) = \6\a (\d\ < n). Let /ibe apositive Borel measure on A. Then the following two conditions are equivalent.

(i) The differentiation operator D defined by Df = f maps Hp{w0) continuously intoL*(d/i).

(ii) There exists a positive constant C such that

u{S(I)) ^ C\I\P I wa(eie) dd, for every interval I cz T. (1-16)

2. Necessary conditions

Before embarking on the proof of Theorem 1 let us introduce some notation. Givenae A, a =t= 0, we define /„ as the open interval in T of length 1 — \a\ whose centre isaj\a\. Also, if A > 0 and / is any open interval in T with A\I\ ^ 2n, AI will denote theopen interval concentric with / whose length is A\I\; if A\I\ > 2n, we set AI = T.

Proof of Theorem 1. Let 1 ^ p < oo, wsAp and suppose that fi is a positive Borel


measure on A such that the differentiation operator D maps Hp(w) continuously intoLp(d/j,), i.e. there exists a positive constant M such that

>. forevery feH?(w). (2-1)

Let I c T be an open interval with |/| < \. We associate to / the point a = a(I) sochosen that a/\a\ is the centre of/ and that 1 — \a\ = |/|, i.e. a is such that / = Ia. Set

Since weAp, it satisfies the doubling condition (see e.g. [12]), that is, there exists a-positive constant B such that

w(eie)dd^B\ w(eie)dd, for all intervals / c T. (2-3)J 21 J I

Setj _ 2*/ jfc = 0 1 m (2"4)

where m is the last integer such that 2m|/| < 2n, and

Im+x = T- (2-5)

It is easy to see that, if a = \a\ e1^ then

' ' ""' ' 3 —n. \6-$\<n.

(Indeed, observing that \a\ > | the estimate is trivial for \6 — <j)\e[Ti/2,n); while if| 0 - 0 | e[0,7r/2) we have

2|sin (0 — 0)| zZ —\6 — 0|

77

and, hence

and the result follows.)Now we easily obtain

6)\^y if e"e/0, (2-6)

and \g(ei0)\^-JL-, if eieelk+1\lk, k = O,l,...,m. (2-7)

Take a natural number n so big tha t

where £ is the constant which appears in (2-3) and define

f(z)=g{z)», N ^ l . (2-9)

156 D A N I E L G I R E L A , M A R I A L O R E N T E AND MARIA D O L O R E S SARRION

I t is clear tha,tfeHp(w) and using (2-6), (2-7), (2-4) and (2-3) we obtain

1 C= Tr

m \ C\g(eie)\nvw(eie)d6+ 2 TT We)\np w(eie) d6

Notice that (2-8) shows that if we define

fc-l

then 4̂ < oo and, hence we have shown

Now we turn to estimate H/'llfp^) from below. The geometric argument used in[3, p. 157] shows that for a = \a\e^ and zeS(I) we have

1-ev—z\a\

which implies \g(z)\> ^rj;, for all zeS(I).o\l\

Now, since |a| > 1/2,

|/'(z)| = ^ ( l - a z ) " " - 1 ! ^ ||gr(2)|n+1, for all

which, with (2-12) and (2-1), implies

Then (2-11) and (2-13) show that, setting

(2-12)

( 2 . 1 3 )

we have

-

1\I\p \ w(eie)d0, for every interval / c T with |/| < \. (2-14)


Now it remains to consider intervals / with |7| ^ §. Notice that, taking/(z) = z in(2-1), we obtain

M C^ (A)<— w(eie)d0. (2-15)

ZnJT

Let / c T be an interval with |/| ^ |, then T = 24/ and then, using (2-15) and (2-3), wehave

p(A) ^ ^ f w(eie) dd = %-[ w{e™) d62nJ 2nj

fM C MR49P C

~B'\ „,(«,«)«*<? < = ^ - [ / Hp I w(eie)dd

which, together with (2-14), proves (1-12) with C = max (Cx, l/2nMB42p). Thisfinishes the proof of Theorem 1.

3. Sufficient conditions

In this section we shall prove Theorem 2 and Theorem 3 with arguments relatedto those used in the proof of [5, theorem 2]. We start with the following lemma.

LEMMA 1. Let 1 < p < oo, and let wbe a non-negative measurable function defined onT which satisfies the doubling condition. Let p: (0,277]^[0, oo) be a bounded Dinifunction. Let jibe a positive Borel measure on A and suppose that there exists a positiveconstant C such that /i satisfies (1"14). Let v be the positive Borel measure on A definedby

Then there exists a positive constant A such that

w(eu)dt, for every interval I cz T. (3-2)

Proof of Lemma 1. Let 1 < p < oo and weAp and let p, fi and v be as in Lemma1. Since p is a bounded Dini function, we see that there exists a positive constant Ksuch that r

Jo

dt<K < oo, (3-3)l o *•

and p(t) <K, if 0 < t < 2n. (3-4)

Let / c T be an interval with |/| ^ 1. For simplicity, say that |/| = h and

/ = leit ;6<t<

We have

^S{1)) = f f n 1 | ,n* < W = P T A P ~ V W dX, (3-5)JJs(/> \l ~\z\) Jo

where Ex = I z e 8(1): —^-r- > A1.

158 D A N I E L GIRELA, MARIA LORENTE AND MARIA DOLORES SARRION

Notice that Ex = 8(1) if 0 < A < hrx and, then (1-14) implies

w(eu)dt, if O<A<A-X. (3-6)

Now, if A ^ h~l, then

EA = {z = reu:l-l/A<r<l, d<t<6+h}. (3-7)

Let m be the natural number such that m ^ Ah < m +1. Then we have

m

Ic(Jl• c 47, (3-8)

where, /, = \eu: 0 + {^ t^ 8+J-^~\, j = 0,1, ...,m.[A A )

Notice that (3-7), (3-8) and the definition of the intervals / ; imply thatEA

and then, using (1-14) and (3-8), we deduce

^ C S l- |̂p/o(|/̂ |) | w(elt)dt = C—/)( Y) S I w(elt)dt3-0

and then, using the fact that w satisfies the doubling condition, i.e. w satisfies (2*3)for some absolute constant B ^ 1, we deduce that

~p\^) \ w(eu)dt, if hr1 < A < oo. (3-9)

Now (3-5), (3-6) and (3-9) imply

w(eu)dt+B2Cp( T T / » ( T ) ^ f w(eu)dt

= Cp(h) I w(eu)dt+B2Cp( I ^f-dt\ f w(eu)dt

which, using (3-3) and (3-4), shows that, setting E = CK(1+B2p), we have

v(S(I)) ^E I w(eu)dt, for every interval / with |/| ^ 1. (3-10)

Notice that, taking / = T in (1'14), we deduce that fi(A) < oo and then with anargument similar to that used to prove (3-10), we obtain v(A) < oo and so there existsa constant y > 0 such that

i>(A) = y I w(eu)dt. (3.11)JT


Now if/ is an interval with |/| > 1, using (3-11) and (2-3), we obtain

C C Cv(S(I)) s; v(A) = y w(eu)dt = y w(elt)dt < yB3 w(ea)dt.

JT J23/ Ji

This and (3-10) prove (3"2) with A = max(£,y53). This finishes the proof ofLemma 1.

Proof of Theorem 2. Let 1 < p < 00 and weAp and let p and fi be as in Theorem2. Let v be defined as in Lemma 1. ~LetfeHp(w). Since Hv(w) c H1, we have (see e.g.[3, theorem 3-6])

and hence

where u(z) denotes the Poisson integral of \f{ei6)\.In view of (3-12), Theorem 2 will be proved if we can show that the operator S

defined by

with u^ denoting the Poisson integral of <j>, maps Lv(w) continuously into Lp(dfi),i.e. there exists a positive constant C such that

, forevery 0eL'(t»). (3-14)

In order to prove (3-14), let us notice that, since weAp, it satisfies the doublingcondition and hence using Lemma 1, we deduce that v satisfies (3-2). Then TheoremB shows that there exists a positive constant C such that

H\\lvm^C\\4>\\hw, forevery <j>eL*(w)

and notice that, by the definition of v, this is equivalent to (3-14). This finishes theproof of Theorem 2.

Proof of Theorem 3. Let weA-^, e > 0, C > 0 and let / tbea positive Borel measurewhich satisfies (1-15). Take 1 < p < 1 + e and let 8 > 0 be defined by p + S = 1+e.Notice that, since weAv it follows that weAp and (1-15) can be written as

fi(S(I)) ^ C\I\P+S I w{ew) dd, for every interval / c T,

and then, using Theorem 2, we deduce that there exists a positive constant C suchthat

g'eL*{d/i), and W\\L>1W^CMH>'M, forevery geH*{w). (3-15)

Let/e//x(w)) and assume t h a t / has no zeros in A. Set g = f1/p. Then

and ||g||&P(I0)=||/IU.(w). (3-16)

160 DANIEL GIRELA, MARIA LORENTE AND MARIA DOLORES SARRION

Let q be the exponent conjugate to p, i.e. q = p/(p—i). Since/' — pq'gv~x, usingHolder's inequality, we obtain

r(z)\d/i(z) ^pWg'h^Jglll^y (3-17)

It is clear that (1*15) and Theorem B imply that there exists a positive constant C1

independent of/ such that

and then, using (3-15) and (3-16), we see that there exists a positive constant C2 suchthat

JJ>((z)\dii(z) ^ Cs\\g\\faw> = ^ l l / I U w (3-18)

Finally iifeH1(w) is arbitrary, then (see [3, theorem 2-5])/ can be factored in theform/(z) = B(z) g(z) where B is a Blaschke product and g is an Hl function which doesnot vanish in A. Setting

jx=q and f2 = (B-

we have that jx and /2 are H1 functions which do not vanish in A and

f = fi+U (3-19)

Now, since \B(ei6)\ = 1, a.e., we have \f{eie)\ = \g(eie)\ a.e., and hence

U = geH\w), and \\h\\HHw)= \\f\\HHw). (3-20)

Also, since \B\ ^ 1, we see that |/2| < 2|/ | and so

f2eH\w), and ||/21|„•<„,< 21|/||„•<„,. (3-21)

Then, since f1 and/2 are Hx{w) functions which do not vanish in A, (3*18), (3-20) and(3-21) show that

which, with (3-19), implies

This finishes the proof.

4. The case 2 ^ p < oo and w(eie) = \6f

This section is devoted to proving Theorem 4. First of all, let us notice that wealready know that (i) => (ii) and so we only have to prove the other implication.Also, we may assume that a =f= 0 since the case a = 0 is covered by theShirokov-Luecking theorem.

So let 2 =$ p < oo and —1 < a < p—1, a 4= 0 and let /tbea positive Borel measureon A. Since there are positive constants A1,A2 such that

A^dl* ^ | 1— eie\" ^A2\6\a, for all de{—n,n), (4-1)


it is clear that the condition (ii) of Theorem 4 is equivalent to the existence of apositive constant C such that

r/i(S(I))^C\I\p \ \l-ei0\add, for every interval / c: T. (4-2)

Ji

Also, it follows from (4-1) that condition (i) of Theorem 4 holds if and only if thedifferentiation operator D maps Hp(\l— ei0\a) continuously into Lp(dji).

Consequently, Theorem 4 will follow from the following.

PROPOSITION 1. Let 2 ^ p < oo and — 1 < a < p— 1, a =j= 0 and let /i be a positive

Borel measure on A which satisfies (1-16) (or, equivalently, (4-2)) for some positiveconstant C. Then the differentiation operator D defined by Df = / ' maps Hp(\l—et6\a)continuously into Lp(d/i).

The proof of Proposition 1 will be based on the following result.

PROPOSITION 2. Let 2 =% p < oo and —1 < a < p— 1, <x=|=0 and let jibe a positiveBorel measure on A which satisfies (1-16) (or, equivalently, (4-2)) for some positiveconstant C. Let v1 and v2 be the positive Borel measures on A defined by

and dv2(z) = d/i(z). (4-4)

Then there exist two positive constants G1 and C2 such that

j>!($(/)) ^ CX\I\, for every interval I a T, (4-5)

and v2(8(I))^C2\I\p+1, for every interval / c T . (4-6)

Proof of Proposition 1. Take p, a and /i as in Proposition 1 and let v1 and v2 bedenned by (4-3) and (4-4) respectively. Proposition 2 states that vx is a Carlesonmeasure and then, using Carleson's theorem, we deduce that Hv is continuouslycontained in Lp(dv1), i.e. there exists a positive constant A1 such that

\g(eie)\pdd, for every geHp. (4-7)

On the other hand, Theorem A and (4-6) imply that the differentiation operator Dmaps Hp continuously into Lp(dv2), i.e. there exists a positive constant A2 such that

\ \

ie\

\g(eie)\pd6, for every geH*. (4-8)

Now \etfeHp(\\-eie\a) and define g(z) =f(z) (l-z)a /p . Then it is clear that

geHp and \\f\\Hp<ii-e">n =

IMIHP- (4'9)We have

f'(z) = -°~

PSP 116


and, hence (since \a/p\ < 1),

\lIV / fr \n<7\\P VIP / CC \n't?\\P \VP

which, using (4-7), (4-8) and (4-9), implies

This proves Proposition 1.

Now it only remains to prove Proposition 2. In order to do so, we shall use theconvention that C will denote a positive constant which may depend on a, p and ftbut is independent of h and 6 and which may not be the same at different occurrences.Also, it is convenient to introduce the following notation. Given #eR and 0 < h ^ 1,we set

SOih = {z = r e u : l-h^r < 1,6 < t<6 + h } , (4-10)and Slh = {z = relt: l-h^r<l,d-h<t<d + h}, (4-11)

Notice t h a t Se< h = 8(1) for / = {eu :6<t<d + h}. Also, if a e C and r > 0, A(o, r) willdenote the open disc with centre a and radius r. The following lemma will be used inthe proof of Proposition 2.

LEMMA 2. Let p, a and /i be as in Proposition 2. Then we have

C ^ , */ \6\ <±h and 1/A<5h; (4-12)

Jsg hf\ All,\))^Chp+c'+1, if \6\<4:h and 5h; (4-13)

and there exists an absolute constant A > 1 such that

f and j<\e\- <4'14)Proof of Lemma 2. If |0| < 4A and I/A < 5A, then Sg< h f) A(l, I/A) c 5* 1/A and then

using (1-16) we obtain

g,h n

This proves (4-12).If |(9| < 4A and 1/A^5h then S M n A(l, I/A) c: fif* sft and, hence

which gives (4-13).Finally, a simple geometric argument shows that there exists an absolute

constant A > 1 such that

if 4 / ^ | 0 K | and j<\0\. (4-15)


This implies (4-14) finishing the proof of Lemma 2.

Proof of Proposition 2. First suppose that 0 < a < p — 1. Take /? with a =% /? ^ p + a

and let /^ be the Borel measure on A defined by

dftp(z) = .fidfi(z). (4-16)

Notice that j ^ = /ia+p and v2 = /*a and then it is easy to see that (4-5) and (4-6) followfrom the following result.

Rl . There exist e > 0 and y > 0 (depending on /?) such that

, */ |0|<f ««d 0<^<e. (4-17)

Proof ofRl. Take |(9| < n/2 and 0 < h< 1/5. We have

J,»: jJ_f >

JA. (4-18)

Consequently, since 0 < a < /̂ ̂ p + oc < p + a + i, using (4-12) and (4-13), we

obtain, for \8\ s$ 77/2 and 0 < h < 1/5

A^dA + yffC ^T±J^C^+-"+ 1 , \d\<4h.Jl/(5A) A

(4-19)

Now, with 4̂ being the constant which appears in Lemma 2, using (116) we obtain,for 4h<8^ n/2 and 0 < 4/A <0 + h

6ih n

If Ah < d sS TT/2 and (9 + A ^ ^ / A , we have

4 + 4 - 1 = C%p+1A-a. (4-20)A 4A/

P I <"*g h fl A ( 1 ^ ) ) < MSe,h) ^ ^ P I <"* ^ CAp+1(6» + A)a. (4-21)

Then, since/? ^ a, (4-18), (4-14), (4-20) and (4-21) show that, for ih <0 s^n/2 we have

fA/(6+h) r{A/0)

A/(6+h)

fA/(6+h) rSg, h) < ChP+1(0 + h)' A^"1 dX + Ch*+1

Jo Jr(A/e) <

J ^/(fl+A) A

0 8 + h


This proves (4-17) for 4ft < 6 ^ n/2. A similar argument can be used to obtain thesame result for —n/2 ^ d < —4ft. This and (4-19) prove Rl with e = 1/5.

Now it remains to prove Proposition 2 in the case — 1 < a < 0. So, let p, ju, and abe as in Proposition 2 with — 1 < a < 0 and let vx and v2 be the Borel measures onA defined by (4-3) and (4-4) respectively. Then just as in the previous case, it is clearthat (4-5) and (4-6) will follow from the following result.

R2 . There exist e > 0 and y > 0 such that

vi(se,h)^7h, if l # l < ^ and 0 < ft < e, (4-22)

and v2(Sgjl)^yhp+\ if \6\ ^ | and 0 < ft < e. (4-23)

Proof of R2. Define v = a + p and S = —a. Then TJ > 0 and 8 > 0 and

(4-24)

We have

Using (4-12), (4-13) and the definition of r\, we obtain for \6\ ^ TT/2 and 0 < h < 1/5

A^dA + ^C ^ - ^ U C A , \d\<4h. (4-25)

If 4ft < (9 ̂ n/2 and 0 ^ 4/A, we have, using (1-16),

^ Chp+1d* ^ Chp+\4:hy = Chp+l+a =

which, with (4-14), show that for 4A < 6 < n/2 we have

6 _ ^ CAvflV+i_±_ = Ch

This proves (4-22) for 4ft < 6 ^ n/2. A similar argument can be used to obtain thesame result for -n/2 «$ d < -4ft. This and (4-25) prove (4-22) with e = 1/5.

Now it remains to prove (4-23). Take \6\ < n/2 and 0 < ft < 1. We have

(4-26)fo

It is clear that there exists a positive constant B such that

S S i 4 c A ( U ) , if \d\ < 2ft and Bh < A,and, hence,

/t(S(,ift\A(l,A)) = 0, if |(9|<2ft and 5ft < A. (4-27)


If|0| < 2h and A ^Bh, using (1-16) and the trivial fact that -Sh < 0 < 6 + h< 3h, weobtain

\, A)) sS fi{8e,h) **Ch?\ Pdt^ 2Chp \ Pdt = Chp+1+° = Chp+1~s,Je Jo

which, with (4-26) and (4-27), shows that

CBh

Vs(Sg,h) < Chp+1~s A'^dA = C^ + 1 , |0| < 2h. (4-28)Jo

Again, a simple geometric argument shows that there exists a positive constantM > 1 such that if 2h < 6 < n/2 and A >M(d + h) then Se>ft\A(l, A) = 0 and, hence,

)) = O, if 2h<6<n/2 and A>J/((9 + ^). (4-29)

Now, if 2h< 6 < n/2 and A ̂ M{d + h), we have

+n

ft\A(l, A)) ^ fi(Se h) ^ Chp tadt^ ChP+W = Chp+1d~s,J

which, with (4-29) and (4-26), implies

vt(Se h) ^ Ch*+Id-S A*"1 dX = CW+1 - ^Jo V d I

This proves (4-23) for 2h < 6 < n/2. With a similar argument, the same result canbe proved for -n/2 < 6 < -2h. This and (4-28) give (4-23). This finishes the proof ofR2.

5. Higher derivatives

Even though so far we have restricted ourselves to the question of characterizingthose positive Borel measures /*onA for which the first derivative/' of any functionfeHp(w) belongs to Lv(dfi) the same question can be studied for higher derivatives.For k = 1,2,3,... and/ analytic in A let Dkf denote the &th-derivative of/. We havethe following results.

THEOREM 5. Let k^l,l^p<co and weAp. Let jibe a positive Borel measure onA. / / the operator Dk maps Hp(w) continuously into Lp(d/i) then there exists a positiveconstant C such that

fi(S(I)) ^ C\I\kp w(eie) dd, for every interval IcT.J i

THEOREM 6. Let k~^l, 1 < p < oo and weAp and let p: (0, 2TT]->[0, oo) be abounded Dini function. Let /J, be a positive Borel measure on A and suppose that thereexists a positive constant C such that

fi(S(l)) ==£ C\I\kpp(\I\) I w{eie) dO, for every interval I c T.

Then the operator Dk maps Hp(w) continuously into Lp(dfi).

166 DANIEL, GIRELA, MARIA LORENTE AND MARIA DOLORES SARRION

THEOREM 7. Let h*^- l,weA1 and let /i be a positive Borel measure on A. Suppose thatthere exist e > 0 and a positive constant C such that

fi(S(I)) ^ C|/|fc+e w(ew)dd, for every interval I c T.

Then the operator Dk maps //*(«;) continuously into L1(d/i).

T H E O R E M 8. Let & 3s 1, 2 ^p < oo, — 1 < a < p— 1 awrf wa(eie) = |0|a (|0| < 77). Le<

[i be a positive Borel measure on A. Then the following two conditions are equivalent, (i)The operator Dk maps Hv(wa) continuously into Lp(d/i). (ii) There exists a positiveconstant C such that

f: C\I\kp wa(e

ie) dd, for every interval I c T.J1

These results can be proved with the ideas used in the proof of the correspondingresults for k = 1 and we shall omit the details. As concerns the proof of Theorem 5let us just say that replacing the estimate | f'(z)\ Js n/2\g(z)\n+1 by the estimate\D)cf(z)\ ̂ Gnk\ g(z)\n+k the proof can be completed in the same way as that ofTheorem 1.

Concerning the proof of Theorem 6 the estimate \Dkf(z)\ ^ [1/(1 —\z\)k]u(z) is usedinstead of (3-12). Then the proof can be finished in the same way as that of Theorem2, applying Lemma 1 with kp instead of p.

The proof of Theorem 7 is a slight modification of that of Theorem 3.Finally, using the analogue of Theorem A for higher order derivatives (see [8,

p. 615]), the proof of Theorem 8 can be completed in the same way as that ofTheorem 4.

We wish to thank the referee for his helpful suggestions for improvement.

REFERENCES[1] L. CARLESON. An interpolation problem for bounded analytic functions. Amer. J. Math. 80

(1958), 921-930.[2] L. CARLESON. Interpolation by bounded analytic functions and the corona problem. Ann. of

Math. 76 (1962), 547-559.[3] P. L. DURBN. Theory of Hv spaces (Academic Press, 1970).[4] J. GAECI'A-CUBRVA and J. L. RUBIO DE FRANCIA. Weighted norm inequalities and related topics

(North-Holland, 1985).[5] D. GIRELA. Mean growth of the derivative of certain classes of analytic functions. Math. Proc.

Cambridge Philos. Soc. 112 (1992), 335-342.[6] D. GIRELA. Growth of the derivative of bounded analytic functions. Complex Variables Theory

Appl. 20 (1992), 221-227.[7] R. HUNT, B. MUCKENHOUPT and R. WHEEDEN. Weighted norm inequalities for the conjugate

function and Hilbert transform. Trans. Amer. Math. Soc. 176 (1973), 227-251.[8] D. H. LUECKING. Embedding derivatives of Hardy spaces into Lebesgue spaces. Proc. London

Math. Soc. (3) 63 (1991), 595-619.[9] J. D. MCPHAIL. A weighted interpolation problem for analytic functions. Studia Math. 96

(1990), 105-116.[10] B. MUCKENHOUPT. Weighted norm inequalities for the Hardy maximal function. Trans.

Amer. Math. Soc. 165 (1972), 207-226.[11] N. A. SHIROKOV. Some generalizations of the Littlewood-Paley Theorem. J. Soviet Math. 8

(1976), 119-129.[12] A. TORCHINSKY. Real-variable methods in harmonic analysis (Academic Press, 1986).

embedding derivatives of weighted hardy spaces into lebesgue spaces

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