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 into is given and the Abel type theorem for mentioned Hardy spaces is proved.

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.

Let denote the space of analytic function on the unit disc . Given spaces , of sequences modelled on , an element is a multiplier of and if for all . We study multipliers between spaces of analytic functions , identifying with a space a space consisting of Taylor’s coefficients of functions from ; that is, It should be pointed out that in general the description of is a considerable challenging quest even for the classical spaces , for example, Hardy spaces (see [3]). While it is clear that , the characterization of is unknown if , . As a matter of fact, there exists a nice description of in terms of convolution operator; see Schur’s theorem in [4]. Note also the famous Hausdorff-Young inequality, which adjudicates that whenever and . We also recall (after [5]) that when restricted to the nonnegative sequences satisfying , then

Multipliers when considered between spaces of analytic functions are often called Hadamard multipliers (or Hadamard products); that is, given spaces of analytic functions and on the Hadamard multipliers of and are defined aswhere is given by , , . 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 , . 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 . 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 be a complete -finite measure space and let denote the space of real valued measurable functions on with the topology of convergence in measure on -finite sets. The order means that for -almost all . If a real Banach space is such that there exists with -a.e. on and with and implies with , then is said to be a Banach lattice (on or on ). If in addition, contains, along with a function , every function equimeasurable with , , then we say that is rearrangement invariant (r.i. space for short).

Throughout the paper we will consider complex r.i. spaces. The term complex r.i. space refers to the complexification of a real r.i. space; that is, if denotes the (real) r.i. space, the complexification of is the Banach space of all complex valued measurable functions on such that the element defined by for is in and . 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 . An (real) r.i. function space on is said to be order continuous if every nonnegative nonincreasing sequence in which converges a.e. to converges to in the norm topology of . If is an order continuous r.i. space on , then the dual space can be identified in a natural way with the Köthe dual space of all such that , for all . An r.i. space is said to be maximal (or to have the Fatou property), if, for any sequence of nonnegative elements from such that for and , one has and .

In the sequel we will use the well-known concept of Boyd indices. Recall that for an r.i. space on we define dilation operators , , by for all and for . Boyd indices and are then defined byIt follows that and . Moreover and (see [7, p. 131]).

Let be a complex r.i. space on . Denote by the space of all such that , where is given by , . The space becomes a Banach space if equipped with the normNote that the spaces are abstract variants of the classical Hardy spaces on the disc, (see [3]), as well as other important spaces of analytic functions like Hardy-Orlicz, Hardy-Lorentz, and Hardy-Marcinkiewicz spaces; see, for example, [810] 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 that the radial limit of is given by provided that the limit exists for almost all . By the lemma of Fatou it follows that if , then exists a.e. on . Here and beneath, for a function , we will write for th Taylor’s coefficient of , . It is clear that the series is convergent uniformly on compact subsets of to the function . For an r.i. space we use the symbol to denote the subspace of elements from consisting of radial limits of functions. It was shown in [8, Proposition ] that if is a maximal r.i. space on then coincides with the space consisting of functions such that the negative Fourier coefficients of vanish. The mapping appeared to be an isometric isomorphism from into .

The key fact in [1] for the study of Hadamard product of spaces was the so-called Riesz property. Let be a linear projection given byIt was Riesz who proved that extends to the bounded projection on if only (see, e.g., [7]). Let be an r.i. space on . We say that has the Riesz property if, for every with the Fourier coefficients , the function belongs to and there exists a constant such thatIt is easy to see that an r.i. space has the Riesz property if its Boyd indices , satisfy . In fact, much more can be proved. In [11] it was shown that has the Riesz property if and only if . In particular, it follows that an r.i. space has the Riesz property if and only if does.

3. Hadamard Multipliers

We study the Hadamard product of functions from Hardy classes generated by r.i. spaces. Given functions the Hadamard product of and is given by the formulaWe note that by the Cauchy-Hadamard formula implies that .

Let and be abstract Hardy spaces generated by r.i. spaces and , respectively. It is easy to verify that the functional given byis a norm on and we have a continuous inclusion . Combining this with the Closed Graph Theorem easily implies that 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 space is a Banach algebra under the convolution given byThe function which is measurable by Fubini’s Theorem belongs to and satisfies

Lemma 1. Let and let be an r.i. space on . Then the following hold: (i) for every .(ii)If , then for every .

Proof. (i) Given any and the seriesconverges uniformly on . Thus the sequenceis bounded on . Since is a bounded function on we have(ii) Let . From [9, Proposition ] it follows (by ) that and Fourier coefficients of satisfy for and for Combining the above proof of (i) with the Lebesgue Domination Theorem yields the second formulae.

In what follows denotes the disc algebra, that is, the space of functions such that extends continuously to the closure of .

Corollary 2. Let , , and be r.i. spaces on . Then the inclusion holds if and only if the restriction of the convolution operator to is bounded from to ; that is, there exists such that for all .

Proof.
Sufficiency. By the Closed Graph Theorem we conclude that there exists such that for every . This implies thatand so, applying Lemma 1, we obtain the required statement on boundedness of the restriction of the convolution operator from to .
Necessity. Fix . Combining Lemma 1 with the boundedness of the convolution operator yields that there exists such that for all and all we haveHenceand so the continuous inclusion follows.

Here comes the main theorem of this section.

Theorem 3. Let , , and be maximal r.i. spaces on . Assume that or is separable. Then the inclusion holds if and only if the convolution operator is bounded from to .

Proof. For any and we haveAssume that . Then the Closed Graph Theorem implies that there exists such that for every . Combining the above we conclude that for all and Since and by Lemma 1(ii) we obtainWe will use the Mean Convergence Theorem from [9, Theorem ] from which it follows (note that is maximal, by assumption) that for every Take any sequence such that . Then (25) implies that there exist and a subsequence of such thatNow observe that for almost all Since and it follows thatand so inequality (27) in combination with the Lebesgue Domination Theorem yieldsThe Fatou property of and inequality (24) implyand this gives the assertion. The reverse implication follows from Corollary 2.

The following result gives a description of the Hadamard multipliers from Hardy spaces into under some mild assumption on an r.i. space .

Theorem 4. Let be a maximal r.i. space on with the Boyd indices satisfying . Then

Proof. Since the convolution operator is bounded from into withit follows from Corollary 2 thatTo prove the reverse continuous inclusion we observe that our hypothesis on the Boyd indices implies that there exists a Riesz projection (see [11]). Thus there exists a constant such thatLet . We claim that . To see this note that is a maximal r.i. space and in consequence translation invariant. Since it follows that for every and Combining inequality (35) with and for all , we conclude that for every This implies by Lemma 1(ii) thatThis proves the claim that and sowith continuous inclusion.

4. Applications

In the following we apply the former theorem to the study of the Abel duals of Hardy spaces. Recall that the Abel dual of a space consists of all such that the limitexists for all (see [12, p. 1223]). We prove the version of the identification of the Abel dual of abstract Hardy spaces in the following.

Theorem 5. Let be an order continuous and maximal r.i. space on and assume that has the Riesz property. The following statements are equivalent. (i)If the function can be represented in the formthen for every .(ii)For any sequence of complex numbers and any , the following limit exists a.e. on :

Proof. (i)(ii). From Theorem 4 it follows that and since has the Riesz property, is a projection of some . Observe that for and the following equality holds:Hölder inequality yieldsSince is maximal, then from the Mean Convergence Theorem (see [9, Theorem ]) it follows that(ii)(i). Take and consider a function . Since , . From (ii) it follows thatexists for almost all and henceconverges for . ThusTake arbitrary and notice that by (ii)exists a.e. on . By the principle of uniform boundedness applied to the family of operators on , it follows that there exists a constant such thatand hence .

Further the description of the dual space of will be needed. Recall that if is maximal r.i. spaces, then with equality of norms and thus is a closed subspace of . As in the proof for the case , , to represent the dual of , it is sufficient to identify the annihilator of in . The proof of the forthcoming lemma is standard and follows the steps of the proof of [3, Theorem ]; nonetheless we include it here for the sake of completeness. The symbol stands for the set of all such that .

Lemma 6. Let be an order continuous and maximal r.i. space on . Then the dual is isometrically isomorphic to . If in addition has the Riesz property, then, for all , there exists a unique function such that

Proof. Assume that is an order continuous and maximal r.i. space on . Every linear and bounded functional on can be represented in the formfor some such that . We will describe the annihilator of in . Take such thatSince polynomials are dense in , it follows that for all and then and . Now, by the assumption is maximal and thus we have .
Conversely, assume that . Then by the Cauchy formulae it follows that for every . Hence is the annihilator of and it follows that isometrically.
Take and observe that by the Hahn-Banach theorem extends to a functional on and hence can be represented in form (52) by a function . This extension is not unique; however, it becomes so if we distinguish in each coset a function for which , . In other wordsNote that has the Riesz property since does and in consequence the analytic projection of belongs to .

In the final theorem of the paper we give a new characterization of belonging to the class in terms of convolution with function.

Theorem 7. Let be an order continuous and maximal r.i. space on and assume that possesses the Riesz property. Forthe following are equivalent: (i)Function belongs to .(ii)For any with the Fourier seriesthe seriesrepresents a function from space .

Proof. (i)(ii). Let us recall that if , , and , then and by [13, Theorem ] there exists a constant such thatIt is easy to see that for . Since has the Riesz property then there exist and such that . Thus two real numbers , satisfying can be found. From Boyd’s theorem (see [14]) it follows that is an interpolation space between and , . Inequality (58) implies that the operator given by is bounded for any . By the interpolation property it follows that is well defined and bounded. Thus for any and the convolution produces a function belonging to . Since is maximal and thus it follows that(ii)(i). Observe that sincefor all with Fourier coefficients , then in particular the seriesis a Fourier series of some . Let us define an operator with the formulaFrom the Closed Graph Theorem it follows that is a bounded operator since in (resp., ) impliesNotice that the adjoint maps into and, by Lemma 6, . From the definition of it follows thatNow take , , . The corresponding is given by and by formula (64) it follows that . Thus is a Hadamard product of and and from Theorem 4 we get .

Competing Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The work was supported by The Foundation for Polish Science (FNP).