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International Journal of Mathematics and Mathematical Sciences
Volume 2011, Article ID 926527, 15 pages
http://dx.doi.org/10.1155/2011/926527
Research Article

-Carleson Measures for a Class of Hardy-Orlicz Spaces

School of Mathematics, Trinity College Dublin, Dublin 2, Ireland

Received 10 December 2010; Accepted 19 April 2011

Academic Editor: Hans Engler

Copyright © 2011 Benoît Florent Sehba. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

An alternative interpretation of a family of weighted Carleson measures is used to characterize -Carleson measures for a class of Hardy-Orlicz spaces admitting a nice weak factorization. As an application, we provide with a characterization of symbols of bounded weighted composition operators and Cesàro-type integral operators from these Hardy-Orlicz spaces to some classical holomorphic function spaces.

1. Introduction

Hardy-Orlicz spaces are the generalization of the usual Hardy spaces. We raise the question of characterizing those positive measures defined on the unit ball of such that these spaces embed continuously into the Lebesgue spaces . More precisely, let denote by the Lebesgue measure on and the normalized measure on the unit sphere which is the boundary of . denotes the space of holomorphic functions on . Let be continuous and nondecreasing function from onto itself. That is, is a growth function. The Hardy-Orlicz space is the space of function in such that the functions , defined by satisfy

We denote the quantity on the left of the above inequality by or simply when there is no ambiguity. Let us remark that , where denotes the Luxembourg (quasi)-norm defined by

Given two growth functions and , we consider the following question. For which positive measures on , the embedding map , is continuous? When and are power functions, such a question has been considered and completely answered in the unit disc and the unit ball in [16]. For more general convex growth functions, an attempt to solve the question appears in [7], in the setting of the unit disc where the authors provided with a necessary condition which is not always sufficient and a sufficient condition. The unit ball version of [7] is given in [8]. To be clear at this stage, let us first introduce some usual notations. For any and , let

These are the higher dimension analogues of Carleson regions. We take as the power functions, that is, for . Thus, the question is now to characterize those positive measures on the unit ball such that there exists a constant such that We call such measures -Carleson measures for . We give a complete answer for a special class of Hardy-Orlicz spaces with , . For simplicity, we denote this space by .

We prove the following result.

Theorem 1.1. Let and . Then the following assertions are equivalent. (i)There exists a constant such that for any and , (i)There exists a constant such that

To prove the above result, we combine weak-factorization results for Hardy-Orlicz spaces (see [9, 10]) and some equivalent characterizations of weighted Carleson measures for which we provide an alternative interpretation. We also provide with some further applications of our characterization of the measures considered here to the boundedness of weighted Cesàro-type integral operators from our Hardy-Orlicz spaces to some holomorphic function spaces in Section 3.

All over the text, , and, , , will denote positive constant not necessarily the same at each occurrence.

This work can be also considered as an application of some recent results obtained by the author and his collaborators [911].

2. -Hardy p-logarithmic Carleson Measures

For and in , we let so that .

Recall that when is a power function, the Hardy-Orlicz space is just the classical Hardy space. More precisely, for , let denote the Hardy space which is the space of all such that We denote by , the space of bounded analytic functions in .

Let be a continuous increasing function from onto itself, and such that for some on for , with . We define the space by where for , the space is the space of polynomials of order in the last coordinates related to an orthonormal basis whose first element is and second element . The integer is taken larger than . For , the quantity appearing in the definition of , we note . The space is then the space of function such that Clearly, coincides with the space of holomorphic functions in such that their boundary values lie in . The space is the usual space of function with bounded mean oscillation while the space of function of logarithmic mean oscillation is given by .

Let denote a positive Borel measure on . The measure is called an -Carleson measure, if there is a finite constant such that for any and any , When , is just called Carleson measure. The infinimum of all these constants will be denoted . We use the notation for . In this section, we are interested in Carleson measure with weights involving the logarithmic function. Let be a positive Borel measure on and . For , a positive function defined on , we say is a -Carleson measure if there is a constant such that for any and , If , is called a -Carleson measure.

We will restrict here to the case , studied by the author in [11] (see also [12] for a special case in one dimension). But here we go beyond the interpretation provided in [11].

2.1. -Hardy -Carleson Measures

In this section, we recall some results of [11] and the notion of -Hardy Carleson measures. We then provide with an alternative interpretation of the results of [11] that will be useful to our characterization. From now on, the notation , where and are two positive constants, will mean there exists an absolute positive constant such that and in this case, we say and are comparable or equivalent. The notation means for some absolute positive constant . Let set

We first recall the following higher dimension version of the theorem of Carleson [1] and its reproducing kernel formulation.

Theorem 2.1. For a positive Borel measure on , and , the following are equivalent (i)The measure is a Carleson measure. (ii)There is a constant such that, for all , (iii)There is a constant such that, for all ,

We note that the constants , in Theorem 2.1 are both comparable to . The proof of this theorem can be found in [13].

We now recall some basic facts about -Hardy measures.

Definition 2.2. Let , and . We say a positive measure on is a -Hardy Carleson measure if there exists a constant such that for all ,

The following high dimension Peter Duren's characterization of -Hardy Carleson measures is useful for our purpose.

Proposition 2.3. Let , and . Let be a positive measure on . Then the following assertions are equivalent.(i)There exists a constant such that for any and any , (ii)There exists a constant such that (iii)There exists a constant such that for all ,

The constants , , and in the above proposition are equivalent. That (i)(ii) can be found in [11]. The equivalence (i)(iii) can be found in [14] for example. We have the following elementary consequence.

Corollary 2.4. Let , , and let be a positive measure on . Then the following assertion are equivalent. (i)There exists a constant such that for any and any , (ii)There exists a constant such that (iii)There exists a constant such that for all , (iv)There exists a constant such that for all and any ,

Proof. The equivalence (i)(ii) is a special case of Proposition 2.3. Note that (iii) is equivalent in saying that for any , the measure is a Carleson measure which is equivalent to (iv). The implication (iv)(i) follows from the usual arguments. Thus, it only remains to prove that (ii)(iii). First by Proposition 2.3, (ii) is equivalent in saying that there exists a constant such that for any , It follows from the hypotheses, the latter, and Hölder's inequality that Thus (ii)(iii). The proof is complete.

Next, we recall the following result proved in [11].

Theorem 2.5. Let , , , and let be a positive Borel measure on . Then the following conditions are equivalent.(i)There is such that for any and , (ii)There is such that (iii)There is such that for any , (iv)There is such that for any , (v)There is such that for any and any ,

Definition 2.6. Let , and . Let be a positive function defined on . We say a positive measure on is a -Hardy -Carleson measure if for any , the measure is a -Carleson measure.

We have the following characterization of -Hardy -Carleson measure which is in fact an alternative interpretation of Theorem 2.5.

Theorem 2.7. Let , , and let be a positive Borel measure on . Then the following conditions are equivalent.(i)There is such that for any and , (ii)There is such that for any , and any , (iii)There is such that for any , and any, (iv)There is such that for any , any , and any , (v)There is such that for any , , and any and ,

Proof. (i)(iv): we first observe with Theorem 2.5 that (i) is equivalent in saying that there is a constant such that for any and any , By Corollary 2.4, the latter is equivalent to (iv).
(ii)(iii)(iv): by rewriting (ii) as where , it follows directly from Theorem 2.5 that (ii)(iii)(iv).
That (iv)(v) is a consequence of Theorem 2.1. The proof is complete.

2.2. -Carleson Measures for Hardy-Orlicz Spaces

In this section, we characterize -Carleson measures of some special Hardy-Orlicz spaces. For this, we will need a weak factorization result of functions in these spaces which follows from the one in [10].

Proposition 2.8. Let . Let denote the Hardy-Orlicz space corresponding to the function . Then the following assertions hold. (i)The product of two functions, one in and the other one in , is in . Moreover, (ii)Any function in the unit ball of admits the following representation (weak factorization): with

Let us remark that the space is the predual of . The following theorem gives a characterization of -Carleson measures of the Hardy-Orlicz spaces considered here.

Theorem 2.9. Let , . Let be the Hardy-Orlicz space corresponding to the function . Then, for a positive measure on , the following assertions are equivalent. (i)There exists a constant such that for any and any , (ii)There exists a constant such that for any ,

Proof. We remark that if (2.38) holds in the unit ball of , then it holds for all . Recall that by Proposition 2.8, every function in the unit ball of weakly factorizes as and . It follows using the equivalent assertion (iv) of Theorem 2.7 that Now we prove that (ii)(i). That (ii) holds implies in particular that for any and any , We observe with Corollary 2.4 that (2.41) is equivalent in saying that for any , the measure is a -Carleson measure or equivalently, By Theorem 2.5, the latter is equivalent to which is equivalent to . The proof is complete.

3. Some Applications

We provide in this section with some applications of -Carleson measures of the above Hardy-Orlicz spaces to the boundedness of multiplication operators, composition operators, and Cesàro integral-type operators. Let us first introduce the generalized Bergman spaces in the unit ball. We recall that for , its radial derivative is the holomorphic function defined by Let , with . The generalized Bergman space consists of holomorphic function such that Clearly, is a Banach under These spaces have been studied in [15]. When , the space corresponds to the usual weighted Bergman space which consists of holomorphic function in such that For and , the corresponding space is just the Hardy space .

Let be a holomorphic function in . We denote by the multiplication operator by defined on by We recall that if is a holomorphic self map of , then the composition operator is defined on by For a holomorphic function in , the weighted composition operator is the composition operator followed by the multiplication by . That is, For a holomorphic function in , the Cesàro-type integral operator is defined by Combining this operator with the weighted composition operator, we obtain a more general operator given by When for all , we write . The multiplication operator, the composition operator, the Cesàro-type integral, and their products have been intensively studied by many authors on various holomorphic function spaces. We refer to the following and the references therein [11, 12, 1630]. As an application of the characterization of -Carleson measures for the Hardy-Orlicz spaces of the previous section, we consider boundedness criteria of the above operators from Hardy-Orlicz spaces to (generalized) weighted Bergman spaces and weighted spaces in the unit ball. We have the following result.

Theorem 3.1. Let , and, . Then is bounded from to if and only if

Proof. Clearly, that is bounded from to is equivalent in saying that there is a constant such that for any , Let us write , . If , then an easy change of variables gives that (3.11) is equivalent to The latter inequality is equivalent in saying that the measure is a -Carleson measure for . It follows from Theorem 2.9 and the equivalent definitions in Theorem 2.7 that (3.11) is equivalent to Changing the variables back, we finally obtain that is bounded from to if and only if The proof is complete.

Remarking that one has we prove in the same way the following result.

Theorem 3.2. Let and with . Then is bounded from to if and only if where .
Let us consider now the operator . We have the following:

Theorem 3.3. Let , , , and . Let . Then is bounded from to , if and only if with .

Proof. We recall that a function is in if and only if the measure is a -Carleson measure (see [31]). That is It follows that is bounded from to if and only if for any , . By the equivalent definition in Theorem 2.7, this is equivalent in saying that for any , , and any , which is equivalent in saying that the measure is a 2-Carleson measure for . It follows from the equivalent definitions of Theorems 2.7 and 2.9 that the latter is equivalent to The proof is complete.

The methods used in this text are quite specific to the case considered here, that is, the embedding . We remark that even in the case , the condition of Theorem 2.9 is still necessary. The proof given here does not allow to say if it is sufficient. In general, the characterization of those positive measures on such that the embedding map ( if and are convex growth functions) is bounded, is still open.

Acknowledgement

The author acknowledges support from the “Irish Research Council for Science, Engineering and Technology”.

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