# Hadamard Multipliers and Abel Dual of Hardy Spaces

Authors: Paweł Mleczko
Journal: Journal of Function Spaces
 (2016)

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## Abstract

The paper is devoted to the study of Hadamard multipliers of functions from the abstract Hardy classes generated by rearrangement invariant spaces. In particular the relation between the existence of such multiplier and the boundedness of the appropriate convolution operator on spaces of measurable functions is presented. As an application, the description of Hadamard multipliers intoH∞ is given and the Abel type theorem for mentioned Hardy spaces is proved.


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## Body

## 1. Introduction

In this paper we analyze so-called Hadamard multipliers between Hardy spaces of analytic functions. Our approach is rather general since we study Hardy spaces generated by rearrangement invariant spaces. The source of inspiration for the current research lies in the paper of Caveny [1] and a recent article of Blasco and Pavlović [2], where the Hadamard multipliers of classical Hardy spaces (see [1]) as well as related notions in more general settings (see [2]) were surveyed.LetH(D) denote the space of analytic function on the unit disc D≔{z∈C:|z|<1}. Given spaces A, B of sequences modelled on N0≔N∪{0}, an element {λn}n∈N0⊂C is amultiplier of A and B if {λnξn}n∈N0∈B for all {ξn}n∈N0∈A. We study multipliers between spaces of analytic functions E=E(D)⊂H(D), identifying with a space E a space E^ consisting of Taylor’s coefficients of functions from E; that is, (1)E^=an:∑n=0∞anzn∈E.It should be pointed out that in general the description of E^ is a considerable challenging quest even for the classical spaces E, for example, Hardy spaces Hp (see [3]). While it is clear that H2^=l2, the characterization of Hp^ is unknown if p≠2, p<∞. As a matter of fact, there exists a nice description of H∞^ in terms of convolution operator; see Schur’s theorem in [4]. Note also the famous Hausdorff-Young inequality, which adjudicates that Hp^⊂lp′ whenever p∈[1,2] and 1/p+1/p′=1. We also recall (after [5]) that when restricted to the nonnegative sequences {an} satisfying an↓0, then(2)an:∑n=0∞anzn∈Hp=an:∑n=0∞n+1p-2anp<∞,p∈1,∞.Multipliers when considered between spaces of analytic functions are often calledHadamard multipliers (orHadamard products); that is, given spaces of analytic functions E and F on D theHadamard multipliers of E and F are defined as(3)E⊙F=∑n=0∞anbnun:an∈E^,bn∈F^,where un:D→C is given by un(z)=zn, z∈D, n∈N0. We refer the reader to [2, 6] for more information and background on this topic.Motivated by the mentioned paper of Caveny [1] we study the Hadamard product between abstract Hardy spaces. Note that in [1] Hadamard multipliers were considered within the settings of Hardy spaces Hp, p∈[1,∞]. We point out that the study of Hadamard product is a well established issue in the theory of spaces of analytic functions (see, e.g., [6]); however, most studies have concentrated on the classical case of Hardy spaces Hp. The purpose of this paper is to extend the research to more general case of Hardy spaces generated by r.i. spaces.


## 2. Rearrangement Invariant and Abstract Hardy Spaces

Let(Ω,Σ,μ) be a complete σ-finite measure space and let L0(Ω)≔L0(Ω,Σ,μ) denote the space of real valued measurable functions on Ω with the topology of convergence in measure on μ-finite sets. The order |x|⩽|y| means that |x(ω)|⩽|y(ω)| for μ-almost all ω∈Ω. If a real Banach space X⊂L0(Ω) is such that there exists u∈X with u>0μ-a.e. on Ω and |x|⩽|y| with x∈L0(Ω) and y∈X implies x∈X with xX⩽yX, then X is said to be aBanach lattice (on Ω or on (Ω,μ)). If in addition, X contains, along with a function f, every function g equimeasurable with f, fX=gX, then we say that X isrearrangement invariant (r.i. space for short).Throughout the paper we will considercomplex r.i. spaces. The termcomplex r.i. space refers to thecomplexification of a real r.i. space; that is, if X denotes the (real) r.i. space, the complexification X(C) of X is the Banach space of all complex valued measurable functions x on Ω such that the element |x| defined by |x|(ω)=|x(ω)| for ω∈Ω is in X and x=xX. For the simplicity of presentation, we will often write r.i. space instead of a complex r.i. space and avoid the use of symbol X(C). An (real) r.i. function space X on (Ω,μ) is said to beorder continuous if every nonnegative nonincreasing sequence in X which converges a.e. to 0 converges to 0 in the norm topology of X. If X is an order continuous r.i. space on (Ω,Σ,μ), then the dual space X∗ can be identified in a natural way with theKöthe dual space (X′,·X′) of all x∈L0(Ω) such that xy∈L1(Ω), for all y∈X. An r.i. space X is said to bemaximal (or to have theFatou property), if, for any sequence (xn) of nonnegative elements from X such that xn↑x for x∈L0(Ω) and sup{xnX:n∈N}<∞, one has x∈X and xnX→xX.In the sequel we will use the well-known concept of Boyd indices. Recall that for an r.i. spaceX on T we define dilation operators Ds:X→X, s>0, by Dsf(t)=f(t/s) for all t⩽2πmin⁡{1,s} and Dsf(t)=0 for 2πs<t⩽2π.Boyd indicespX and qX are then defined by(4)pX=lims→∞⁡log⁡slog⁡Ds,qX=lims→0+⁡log⁡slog⁡Ds.It follows that pX,qX∈[1,∞] and pX⩽qX. Moreover 1/pX+1/pX′=1 and 1/qX+1/qX′=1 (see [7, p. 131]).LetX be a complex r.i. space on T≔[0,2π). Denote by HX the space of all f∈H(D) such that sup{frei·X:r∈[0,1)}<∞, where fr:∂D→C is given by fr(eiθ)=f(reiθ), θ∈T. The space HX becomes a Banach space if equipped with the norm(5)fHX≔sup⁡frei·X:r∈0,1.Note that the spaces HX are abstract variants of the classical Hardy spaces Hp on the disc, p∈[1,∞] (see [3]), as well as other important spaces of analytic functions like Hardy-Orlicz, Hardy-Lorentz, and Hardy-Marcinkiewicz spaces; see, for example, [8–10] for the recent studies on this topic.The study of functions analytic on the disc is intimately connected to the study of the boundary functions. Recall thatthe radial limitf∗ of f∈H(D) is given by f∗(eiθ)=limr→1-fr(eiθ) provided that the limit exists for almost all θ∈T. By the lemma of Fatou it follows that if f=∑n=0∞‍f^(n)un∈HX, then f∗ exists a.e. on T. Here and beneath, for a function f∈H(D), we will write f^(n) for nth Taylor’s coefficient of f, n∈N0. It is clear that the series ∑n=0∞‍f^(n)un is convergent uniformly on compact subsets of D to the function f. For an r.i. space X we use the symbol HX~ to denote the subspace of elements from X consisting of radial limits of HX functions. It was shown in [8, Proposition 2.2] that if X is a maximal r.i. space on T then HX coincides with the space consisting of functions f∈X such that the negative Fourier coefficients of f vanish. The mapping f↦f∗ appeared to be an isometric isomorphism from HX into X.The key fact in [1] for the study of Hadamard product of Hp spaces was the so-called Riesz property. Let R be a linear projection given by(6)R∑k=-nnakekit=∑k=0nakekit,t∈T,n∈N0.It was Riesz who proved that R extends to the bounded projection on Lp(T) if only p∈(1,∞) (see, e.g., [7]). Let X be an r.i. space on T. We say that X has theRiesz property if, for every f∈X with the Fourier coefficients {f^(n)}, the function ∑n=0∞‍f^(n)un belongs to HX and there exists a constant C>0 such that(7)∑n=0∞f^nunHX⩽CfX.It is easy to see that an r.i. space X has the Riesz property if its Boyd indices pX, qX satisfy pX,qX∈(1,∞). In fact, much more can be proved. In [11] it was shown that X has the Riesz property if and only if pX,qX∈(1,∞). In particular, it follows that an r.i. space X has the Riesz property if and only if X′ does.


## 3. Hadamard Multipliers

We study the Hadamard product of functions from Hardy classes generated by r.i. spaces. Given functionsf,g∈H(D) the Hadamard product of f and g is given by the formula(8)f⊙g=∑n=0∞f^ng^nun.We note that by the Cauchy-Hadamard formula f,g∈H(D) implies that f⊙g∈H(D).LetHX and HY be abstract Hardy spaces generated by r.i. spaces X and Y, respectively. It is easy to verify that the functional given by(9)gHX⊙HY=sup⁡f⊙gHY:fHX⩽1is a norm on HX⊙HY and we have a continuous inclusion HX⊙HY↪H(D). Combining this with the Closed Graph Theorem easily implies that HX⊙HY is a Banach space equipped with the norm given in (9).Before we state and prove a technical lemma which will be useful in the sequel we recall that the spaceL1=L1(∂D) is a Banach algebra under the convolution ∗ given by(10)f∗geiθ=12π∫02πfeitgeiθ-tdt,f,g∈L1,θ∈T.The function f∗g which is measurable by Fubini’s Theorem belongs to L1(∂D) and satisfies(11)f∗gL1⩽fL1gL1.Lemma 1.
Letf,g∈H(D) and let X be an r.i. space on T. Then the following hold: (i)
(f⊙g)r=fr∗gr for every r∈[0,1).(ii)
Iff∈HX, then (f⊙g)r=f∗∗gr for every r∈[0,1).Proof.
(i) Given anyr∈[0,1) and θ∈T the series(12)∑n=0∞g^krkeinθ-t,t∈Tconverges uniformly on T. Thus the sequence(13)∑n=0∞g^krkeinθ-tn∈N0is bounded on T. Since fr is a bounded function on ∂D we have(14)fr∗greiθ=12π∫02πfreit∑n=0∞g^nrneinθ-tdt=∑n=0∞g^nrneinθ∫02πfreite-intdt=∑n=0∞f^ng^nreiθn=f⊙greiθ.
(ii) Letf∈HX. From [9, Proposition 2.2] it follows (by X′′⊂L1(T)) that f∗∈L1(∂D) and Fourier coefficients of f∗ satisfy for n<0(15)f∗^n=12π∫02πf∗eite-intdt=0and for n∈N0(16)f∗^n=f^n.Combining the above proof of (i) with the Lebesgue Domination Theorem yields the second formulae.In what followsA(D) denotes the disc algebra, that is, the space of functions f∈H(D) such that f extends continuously to the closure D¯ of D.Corollary 2.
LetX, Y, and Z be r.i. spaces on T. Then the inclusion (17)HZ⊂HX⊙HYholds if and only if the restriction of the convolution operator to A(D)×A(D) is bounded from X×Z to Y; that is, there exists C>0 such that f∗gY⩽CfXgZ for all f,g∈A(D).Proof.
Sufficiency. By the Closed Graph Theorem we conclude that there exists C>0 such that gHX⊙HY⩽CgHZ for every g∈HZ. This implies that(18)f⊙gHY⩽fHXgHZ,f∈HX,g∈HZand so, applying Lemma 1, we obtain the required statement on boundedness of the restriction of the convolution operator from X×Z to Y.
Necessity. Fix g∈HZ. Combining Lemma 1 with the boundedness of the convolution operator yields that there exists C>0 such that for all f∈HX and all r∈[0,1) we have(19)f⊙grY=fr∗grY⩽CfrXgrZ⩽CfHXgHZ.Hence(20)f⊙gHY⩽CfHXgHZand so the continuous inclusion HZ↪HX⊙HY follows.Here comes the main theorem of this section.Theorem 3.
LetX, Y, and Z be maximal r.i. spaces on T. Assume that X or Y is separable. Then the inclusion (21)HZ⊂HX⊙HYholds if and only if the convolution operator is bounded from HX~×HY~ to Z.Proof.
For anyf∈HX and g∈HX⊙HY we have(22)f⊙grY⩽f⊙gHY⩽fHXgHX⊙HY,r∈0,1.Assume that HZ⊂HX⊙HY. Then the Closed Graph Theorem implies that there exists C such that gHX⊙HY⩽CgHZ for every g∈HZ. Combining the above we conclude that for all f∈HX and g∈HZ(23)f⊙grY⩽CfHXgHZ,r∈0,1.Since fHX=fHX~ and gHZ=gHZ~ by Lemma 1(ii) we obtain(24)f∗∗grY⩽CfHX~gHZ~,r∈0,1.We will use the Mean Convergence Theorem from [9, Theorem 3.2] from which it follows (note that Z is maximal, by assumption) that for every g∈HZ(25)limr→1-⁡gr-g∗Z=0.Take any sequence (rn)⊂[0,1) such that rn→1. Then (25) implies that there exist G∈Z and a subsequence (rkn) of (rn) such that(26)grkn⟶g∗a.e.onT,(27)grkn⩽Ga.e.onT.Now observe that for almost all θ,t∈T(28)f∗eitgrkneiθ-t⟶f∗eitg∗eiθ-t.Since f∗∈L1(∂D) and G∈L1(∂D) it follows that(29)12π∫02πf∗eitGeiθ-tdt<∞a.e.onTand so inequality (27) in combination with the Lebesgue Domination Theorem yields(30)f∗∗grkn⟶f∗∗g∗a.e.onT.The Fatou property of Y and inequality (24) imply(31)f∗∗g∗Y⩽liminfn→∞⁡f∗∗grknY⩽Cf∗Xg∗Z=CfHX~gHY~and this gives the assertion. The reverse implication follows from Corollary 2.The following result gives a description of the Hadamard multipliers from Hardy spacesHX into H∞ under some mild assumption on an r.i. space X.Theorem 4.
LetX be a maximal r.i. space on T with the Boyd indices satisfying pX,qX∈(1,∞). Then(32)HX⊙H∞=HX′.Proof.
Since the convolution operator is bounded fromX×X′ into L∞ with(33)f∗gL∞⩽fXgX′,f,g∈X×X′it follows from Corollary 2 that(34)HX′↪HX⊙H∞.To prove the reverse continuous inclusion we observe that our hypothesis on the Boyd indices implies that there exists a Riesz projection R:X→HX~ (see [11]). Thus there exists a constant C=CX⩾0 such that(35)RFHX~⩽CFX,F∈X.
Letg∈HX⊙H∞. We claim that g∈HX′. To see this note that X′ is a maximal r.i. space and in consequence translation invariant. Since L∞(T)⊂X′ it follows that for every r∈[0,1) and θ∈T(36)grX′=sup⁡12π∫02πFeitgreiθ-tdt:FX⩽1.Combining inequality (35) with RX=HX~ and fHX=f∗X for all f∈HX, we conclude that for every θ∈T(37)
g
r
X
′
⩽
C
·
sup
⁡
1
2
π
∫
0
2
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R
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i
t
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r
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i
θ
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t
d
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R
F
X
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1
=
C
·
sup
⁡
1
2
π
∫
0
2
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f
∗
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i
t
g
r
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.This implies by Lemma 1(ii) that(38)grX′⩽Csup⁡f∗greiθ:fHX⩽1,r∈0,1⩽Csup⁡f⊙gH∞:fHX⩽1.This proves the claim that g∈HX′ and so(39)HX⊙H∞↪HX′with continuous inclusion.


## 4. Applications

In the following we apply the former theorem to the study of the Abel duals of Hardy spaces. Recall that theAbel dual of a space X⊂X(D) consists of all g∈H(D) such that the limit(40)limr→1-⁡∑n=0∞f^ng^nrnexists for all f∈X (see [12, p. 1223]). We prove the version of the identification of the Abel dual of abstract Hardy spaces in the following.Theorem 5.
LetX be an order continuous and maximal r.i. space on T and assume that X has the Riesz property. The following statements are equivalent. (i)
If the functiong∈H(D) can be represented in the form(41)g=∑n=0∞g^nunthenf⊙g∈H∞ for every f∈HX.(ii)
For any sequence of complex numbers{λn}n∈N0 and any f∈HX, the following limit exists a.e. on T:(42)limr→1-⁡∑n=0∞f^nλnrneinθ.Proof.
(i)⇒(ii). From Theorem 4 it follows that g∈HX′ and since X′ has the Riesz property, g is a projection of some G∈X′. Observe that for 0⩽r<ρ<1 and 0⩽s<ρ<1 the following equality holds:(43)∫02πftgteiθ-t-gseiθ-tdt=∫02πfr/ρeit-fs/ρeitgρeiθ-tdt.Hölder inequality yields(44)f⊙greiθ-f⊙gseiθ=12π∫02πfr/ρt-fs/ρtgsθ-tdt⩽fr/ρ-fs/ρXgsX′.Since X is maximal, then from the Mean Convergence Theorem (see [9, Theorem 3.2]) it follows that(45)limr→1-⁡∑n=0∞f^nλnrneinθ=limr→1-⁡f⊙greiθa.e.onT.(ii)⇒(i). Take 0<δ<1 and consider a function fδ≔∑n=0∞‍δnun. Since L∞(T)⊂X, fδ∈HX. From (ii) it follows that(46)limr→1-⁡∑n=0∞δnλnrneinθexists for almost all θ∈T and hence(47)∑n=0∞λnrneinθconverges for r∈[0,1). Thus(48)g≔∑n=0∞λnun∈HD.Take arbitrary f∈HX and notice that by (ii)(49)limr→1-⁡12π∫02πf∗eitgreiθ-tdt=limr→1-⁡f⊙greiθexists a.e. on T. By the principle of uniform boundedness applied to the family (f∗gr)r∈[0,1) of operators f↦f∗gr on HX, it follows that there exists a constant C>0 such that(50)12π∫02πfeitgreiθ-tdt⩽CfHX,f∈HXand hence f⊙g∈H∞.Further the description of the dual space ofHX will be needed. Recall that if X is maximal r.i. spaces, then HX=HX~ with equality of norms and thus HX~ is a closed subspace of X. As in the proof for the case X=Lp, p∈[1,∞), to represent the dual of (HX)∗, it is sufficient to identify the annihilator of HX~ in X∗. The proof of the forthcoming lemma is standard and follows the steps of the proof of [3, Theorem 7.1]; nonetheless we include it here for the sake of completeness. The symbol HX0 stands for the set of all f∈HX such that f(0)=0.Lemma 6.
LetX be an order continuous and maximal r.i. space on T. Then the dual X∗ is isometrically isomorphic to X′/HX0′~. If in addition X has the Riesz property, then, for all φ∈(HX)∗, there exists a unique function g∈HX′~ such that(51)φf=12π∫02πf∗eiθg∗eiθ¯dθ,f∈HX.Proof.
Assume thatX is an order continuous and maximal r.i. space on T. Every linear and bounded functional φ on X can be represented in the form(52)φf=12π∫02πfeiθgeiθdθ,for some g∈X′ such that φX∗=gX′. We will describe the annihilator of HX~ in X′. Take g∈X′ such that(53)∫02πgeiθf∗eiθdθ=0∀f∈HX.Since polynomials are dense in HX, it follows that g^(n)=∫T‍g(eiθ)einθdθ=0 for all n∈N0 and then g∈HX′ and g(0)=0. Now, by the assumption X′ is maximal and thus we have g∈HX0′.
Conversely, assume thatg∈HX0′~. Then by the Cauchy formulae it follows that ∫T‍f∗(eiθ)g(eiθ)dθ=0 for every f∈HX. Hence HX0~ is the annihilator of HX~ and it follows that (HX)∗≅X′/HX0′~ isometrically.
Takeφ∈(HX)∗ and observe that by the Hahn-Banach theorem φ extends to a functional on X and hence φ can be represented in form (52) by a function g∈X′. This extension is not unique; however, it becomes so if we distinguish in each coset a function g for which g^(n)=0, n∈N. In other words(54)φf=∫02πf∗eiθgeiθ¯dθ,f∈HX.Note that X′ has the Riesz property since X does and in consequence the analytic projection of g∈X′ belongs to HX′.In the final theorem of the paper we give a new characterization of belonging to the classHX in terms of convolution with L1 function.Theorem 7.
LetX be an order continuous and maximal r.i. space on T and assume that X possesses the Riesz property. For(55)f=∑n=0∞f^nun∈HDthe following are equivalent: (i)
Functionf belongs to HX.(ii)
For anyg∈L1(T) with the Fourier series(56)∑n=-∞∞g^neinθ,θ∈Tthe series(57)∑n=0∞f^ng^nunrepresents a function from spaceHX.Proof.
(i)⇒(ii). Let us recall that if f∈Hp, p∈[1,∞), and g∈L1, then f∗∗g∈Lp and by [13, Theorem 1.15] there exists a constant C>0 such that(58)f∗∗gLp⩽CfHp.It is easy to see that f∗∗g^(n)=f^(n)g^(n) for n∈N0. Since X has the Riesz property then there exist pX and qX such that pX,qX∈(1,∞). Thus two real numbers p, q satisfying 1<p<pX⩽qX<q<∞ can be found. From Boyd’s theorem (see [14]) it follows that X is an interpolation space between Lp and Lq, p,q∈(1,∞). Inequality (58) implies that the operator T:Hp→Lp given by Tf=f∗∗g is bounded for any p∈[1,∞). By the interpolation property it follows that T:HX→X is well defined and bounded. Thus for any f∈HX and g∈L1 the convolution f∗∗g produces a function belonging to X. Since X is maximal and thus X′′=X it follows that(59)∑n=0∞f^ng^nun∈HX.(ii)⇒(i). Observe that since(60)∑n=0∞f^ng^nun∈HXfor all g∈L1 with Fourier coefficients {g^(n)}n=-∞∞, then in particular the series(61)∑n=0∞f^ng^-ne-inθis a Fourier series of some F∈X. Let us define an operator S:L1/H01→X/X0 with the formula(62)Sg+H01=F+HX0,g∈L1.From the Closed Graph Theorem it follows that S is a bounded operator since fn→f in L1/H01 (resp., X/X0) implies(63)limn→∞⁡fn^k=f^k,k∈N0.Notice that the adjoint S∗ maps (X/HX0)∗ into (L1/H01)∗ and, by Lemma 6, S∗:HX′~→H∞. From the definition of S it follows that(64)∫02πh∗eiθFeiθdθ=∫02πS∗heiθgeiθdθ,h∈HX′.Now take g(eiθ)=eikθ, θ∈T, k∈N0. The corresponding F is given by F(eiθ)=f^(k)e-ikθ and by formula (64) it follows that S∗h^(k)=f^(k)h^(k). Thus S∗ is a Hadamard product of f∈HX and h∈HX′ and from Theorem 4 we get f∈HX.



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