Hadamard Multipliers and Abel Dual of Hardy Spaces

The paper is devoted to the study of Hadamardmultipliers 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.


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 (D) denote the space of analytic function on the unit disc D fl { ∈ C : || < 1}.Given spaces ,  of sequences modelled on N 0 fl N ∪ {0}, an element {  } ∈N 0 ⊂ C is a multiplier of  and  if {    } ∈N 0 ∈  for all {  } ∈N 0 ∈ .We study multipliers between spaces of analytic functions  = (D) ⊂ (D), identifying with a space  a space Ê consisting of Taylor's coefficients of functions from ; that is, It should be pointed out that in general the description of Ê is a considerable challenging quest even for the classical spaces , for example, Hardy spaces   (see [3]).While it is clear that Ĥ2 = ℓ 2 , the characterization of Ĥ is unknown if  ̸ = 2,  < ∞.As a matter of fact, there exists a nice description of Ĥ∞ in terms of convolution operator; see Schur's theorem in [4].Note also the famous Hausdorff-Young inequality, which adjudicates that Ĥ ⊂ ℓ   whenever  ∈ [1, 2] and 1/ + 1/  = 1.We also recall (after [5]) that when restricted to the nonnegative sequences {  } satisfying   ↓ 0, 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 D the Hadamard multipliers of  and  are defined as where   : D → C is given by   () =   ,  ∈ D,  ∈ N 0 .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   ,  ∈ [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   .The purpose of this paper is to extend the research to more general case of Hardy spaces generated by r.i.spaces.

2
Journal of Function Spaces

Rearrangement Invariant and Abstract Hardy Spaces
Let (Ω, Σ, ) be a complete -finite measure space and let  0 (Ω) fl  0 (Ω, Σ, ) denote the space of real valued measurable functions on Ω with the topology of convergence in measure on -finite sets.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 (C) of  is the Banach space of all complex valued measurable functions  on Ω such that the element || defined by ||() = |()| for  ∈ Ω 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 (C).An (real) r.i.function space  on (Ω, ) is said to be order continuous if every nonnegative nonincreasing sequence in  which converges a.e. to 0 converges to 0 in the norm topology of .If  is an order continuous r.i.space on (Ω, Σ, ), then the dual space  * can be identified in a natural way with the Köthe dual space (  , ‖⋅‖   ) of all  ∈  0 (Ω) such that  ∈  1 (Ω), 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  ∈  0 (Ω) and sup {‖  ‖  :  ∈ N} < ∞, one has  ∈  and ‖  ‖  → ‖‖  .
In the sequel we will use the well-known concept of Boyd indices.Recall that for an r.i.space  on T we define dilation operators   :  → ,  > 0, by   () = (/) for all  ⩽ 2 min{1, } and   () = 0 for 2 <  ⩽ 2.
Note that the spaces  are abstract variants of the classical Hardy spaces   on the disc,  ∈ [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][9][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 that the radial limit  * of  ∈ (D) is given by  * (  ) = lim →1 −   (  ) provided that the limit exists for almost all  ∈ T. By the lemma of Fatou it follows that if  = ∑ ∞ =0 f()  ∈ , then  * exists a.e. on T.Here and beneath, for a function  ∈ (D), we will write f() for th Taylor's coefficient of ,  ∈ N 0 .It is clear that the series ∑ ∞ =0 f()  is convergent uniformly on compact subsets of D to the function .For an r.i.space  we use the symbol H to denote the subspace of elements from  consisting of radial limits of  functions.It was shown in [8, Proposition 2.2] that if  is a maximal r.i.space on T 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 by  ( It was Riesz who proved that  extends to the bounded projection on   (T) if only  ∈ (1, ∞) (see, e.g., [7]).Let  be an r.i.space on T. We say that  has the Riesz property if, for every  ∈  with the Fourier coefficients { f()}, the function ∑ ∞ =0 f()  belongs to  and there exists a constant  > 0 such that It is easy to see that an r.i.space  has the Riesz property if its Boyd indices   ,   satisfy   ,   ∈ (1, ∞).In fact, much more can be proved.In [11] it was shown that  has the Riesz property if and only if   ,   ∈ (1, ∞).In particular, it follows that an r.i.space  has the Riesz property if and only if   does.

Hadamard Multipliers
We study the Hadamard product of functions from Hardy classes generated by r.i.spaces.Given functions ,  ∈ (D) the Hadamard product of  and  is given by the formula We 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  1 =  1 (D) is a Banach algebra under the convolution * given by The function  *  which is measurable by Fubini's Theorem belongs to  1 (D) and satisfies Lemma 1.Let ,  ∈ (D) and let  be an r.i.space on T.
Then the following hold: Proof.(i) Given any  ∈ [0, 1) and  ∈ T the series converges uniformly on T. Thus the sequence is bounded on T. Since   is a bounded function on D we have Combining the above proof of (i) with the Lebesgue Domination Theorem yields the second formulae.
In what follows (D) denotes the disc algebra, that is, the space of functions  ∈ (D) such that  extends continuously to the closure D of D.
and so the continuous inclusion  →  ⊙  follows.
Here comes the main theorem of this section.
To prove the reverse continuous inclusion we observe that our hypothesis on the Boyd indices implies that there exists a Riesz projection R :  → HX (see [11]).Thus there exists a constant  =   ⩾ 0 such that Let  ∈ ⊙ ∞ .We claim that  ∈   .To see this note that   is a maximal r.i.space and in consequence translation invariant.Since  ∞ (T) ⊂   it follows that for every  ∈ [0, This proves the claim that  ∈   and so with continuous inclusion.

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  ⊂ (D) consists of all  ∈ (D) such that the limit lim exists 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 T and assume that  has the Riesz property.The following statements are equivalent.
(i) If the function  ∈ (D) can be represented in the form then  ⊙  ∈  ∞ for every  ∈ .
(ii) For any sequence of complex numbers {  } ∈N 0 and any  ∈ , the following limit exists a.e. on T: Proof.(i)⇒(ii).From Theorem 4 it follows that  ∈   and since   has the Riesz property,  is a projection of some  ∈   .Observe that for 0 ⩽  <  < 1 and 0 ⩽  <  < 1 the following equality holds: Further the description of the dual space of  will be needed.Recall that if  is maximal r.i.spaces, then  = H with equality of norms and thus H is a closed subspace of .As in the proof for the case  =   ,  ∈ [1, ∞), to represent the dual of () * , it is sufficient to identify the annihilator of H in  * .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  0 stands for the set of all  ∈  such that (0) = 0. Lemma 6.Let  be an order continuous and maximal r.i.space on T. Then the dual  * is isometrically isomorphic to   / H  0 .If in addition  has the Riesz property, then, for all  ∈ () * , there exists a unique function  ∈ H  such that Proof.Assume that  is an order continuous and maximal r.i.space on T. Every linear and bounded functional  on  can be represented in the form for some  ∈   such that ‖‖  * = ‖‖   .We will describe the annihilator of H in   .Take  ∈   such that Since polynomials are dense in , it follows that ĝ() = ∫ T (  )   = 0 for all  ∈ N 0 and then  ∈   and (0) = 0. Now, by the assumption   is maximal and thus we have  ∈   0 .Conversely, assume that  ∈ H  0 .Then by the Cauchy formulae it follows that ∫ T  * (  )(  ) = 0 for every  ∈ .Hence H 0 is the annihilator of H and it follows that () * ≅   / H  0 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 ĝ() = 0,  ∈ N. In other words Note 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  1 function.
Theorem 7. Let  be an order continuous and maximal r.i.space on T and assume that  possesses the Riesz property.For the following are equivalent: (i) Function  belongs to .

Corollary 2 .
Let , , and  be r.i.spaces on T. Then the inclusion is bounded from  ×  to ; that is, there exists  > 0 such that ‖ * ‖  ⩽ ‖‖  ‖‖  for all ,  ∈ (D).By the Closed Graph Theorem we conclude that there exists  > 0 such that ‖‖ ⊙ ⩽ ‖‖  for every  ∈ .
Necessity.Fix  ∈ .Combining Lemma 1 with the boundedness of the convolution operator yields that there exists  > 0 such that for all  ∈  and all  ∈ [0, 1) we have     ( ⊙ )      =       *       ⩽ Since ‖‖  = ‖‖ H and ‖‖  = ‖‖ H by Lemma 1(ii)The following result gives a description of the Hadamard multipliers from Hardy spaces  into  ∞ under some mild assumption on an r.i.space .Let  be a maximal r.i.space on T with the Boyd indices satisfying   ,   ∈ (1, ∞).Then  ⊙  ∞ =   .