Journal of Function Spaces and Applications

Volume 2012, Article ID 792763, 21 pages

http://dx.doi.org/10.1155/2012/792763

## Carleson Measure Theorems for Large Hardy-Orlicz and Bergman-Orlicz Spaces

^{1}Département de Mathématiques, Université Paris-Sud, Bâtiment 425, 91405 Orsay, France^{2}School of Mathematics, Trinity College, Dublin 2, Ireland

Received 24 October 2011; Accepted 20 February 2012

Academic Editor: Miroslav Englis

Copyright © 2012 Stéphane Charpentier and Benoît 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

We characterize those measures for which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) space (resp., ) of the unit ball of embeds boundedly or compactly into the Orlicz space (resp., ), when the defining functions and are growth functions such that for , and such that is nondecreasing. We apply our result to the characterization of the boundedness and compactness of composition operators from (resp., ) into (resp., ).

#### 1. Introduction and Preliminaries

##### 1.1. Introduction

Let and , , denoting, respectively, the unit ball and the unit sphere of . For , we denote by the unit disc of the complex plane.

For a large class of spaces of holomorphic functions in the unit disc or the unit ball, characterizations of the boundedness and compactness of the canonical embedding have been given and applied to different areas, for example, interpolation, multipliers, integral operators, composition operators, and so forth. These results are known as Carleson’s type theorems.

First, when , Carleson [1] proved that if and only if the finite positive Borel measure on (or ) is a so-called *Carleson measure*. This result was extended to the unit ball by Hörmander [2], whose proof was simplified by Power [3]. Duren [4] characterized those measures such that , with , in terms of -*Carleson measures*. For the unweighted and weighted Bergman spaces , , similar results were obtained by Cima and Wogen [5], Luecking [6], Ueki [7]. We recall also that the compactness of was also characterized in the previous cases, in terms of *vanishing* Carleson’s type measures. It is usual to assume that when the measure is defined on , then is absolutely continuous with respect to . This assumption will be done, without mentioning it any further.

Some observations may be done. First, it appears that the characterizations of both boundedness and compactness of are not always satisfied and just depend on the ratio for ; in particular, when , they are independent of . Now, since the restriction to of the finite positive Borel is assumed to be absolutely continuous with respect to the Lebesgue measure, then it is trivial that always holds. On the contrary, the compactness of this inclusion implies strong condition on . This suggests to think about this question when the space is intercalated between every and .

This motivation is reinforced by a second observation: if the measure is the pull-back measure of the invariant-rotation measure on under a holomorphic map , , then is always a Carleson measure when (this is the Littlewood Subordination Principle, see [8]), but it is not systematic for . For the compactness, there is still a big gap between and for any . This observation is directly connected to the study of composition operators on , which are defined by for , and which may be seen as the embedding operators . By the way, this leads the authors of [9, 10] to state Carleson theorems for Hardy-Orlicz and Bergman-Orlicz spaces (resp., denoted by and ) in the unit disc, when the defining function is an *Orlicz function*. These spaces appear as good candidates for generalizing and spaces, , and for covering the gap with . This fact was still more pointed out in [11, 12], where the author gave Carleson theorems in the unit ball, under some mild conditions on the defining function . Indeed, he showed that, when is the pull-back measure under a holomorphic self-map of , then always embeds into whenever satisfied a fast growth condition (namely, the -*Condition*, that implies for large value of and means that we are *close* to ). Similarly, it was shown in [13] that if compactly embeds into for *every *, then compactly (the converse easily holds). We mention that there is no such that if and only if [14], and that such previous results are not true for arbitrary measure . Yet, a link has been made between the involved Carleson conditions and the type of growth of the Orlicz function . This is also strengthened by the fact that if the Orlicz function is *dominated* by a power function (exactly satisfies the -Condition), then boundedly (resp., compactly) if and only if , that is, if and only if is a classical Carleson measure (resp., a vanishing Carleson measure).

Let us note that similar results hold for Bergman-Orlicz spaces.

The purpose of the present paper is to deal with the same kind of question on the opposite side, that is, for Hardy-Orlicz and Bergman-Orlicz spaces which are larger than . It seems that nothing has been done in this direction, except for explicit functions . In particular, the second author gave a necessary and sufficient condition for the inclusion to be bounded, when , , and [15]. Moreover, [16] characterized Carleson measures for , where , , that is, when is the area Nevanlinna space. These measures reveal to be those for which holds, which are Bergman-Carleson measures.

A first difficulty when dealing with large Hardy-Orlicz or Bergman-Orlicz spaces is that we do not have normed spaces any more, and we need to exhibit the *good properties* of the function , in order to define spaces with which it is *reasonable* to work. For that, we were inspired by [17, 18] and the references therein. Then, we obtain more generally a complete characterization of those finite positive Borel measures such that (resp., ) is bounded or compact, when and are two *growth* functions (i.e., for which (resp., ) for some , ), and such that *grows faster* than . It appears that if , then such measures are exactly those which are Carleson (resp., Bergman-Carleson) measures for some . For the Bergman-Orlicz case, these results let one think that, between the area Nevanlinna class [16] and , there is no difference regarding to Carleson theorems, whenever the defining functions share some natural properties. This is in contrast with what happens between and .

The paper is organized as follows. In the next subsection, we introduce the Hardy-Orlicz and Bergman-Orlicz spaces under further considerations, proving or generalizing some useful and classical results. The second section consists of the statements and the proofs of our Carleson theorems for these spaces. A third and last part is an immediate application of our result to composition operators.

*Notation*. Given two points , the euclidean inner product of and will be denoted by , that is, ; the notation will stand for the associated norm, as well as for the modulus of a complex number.

will stand for the invariant-rotation measure on the unit sphere. For will be the measure on defined by , where is the Lebesgue measure on and is the constant of normalization.

We will use the notations and for one-sided estimates up to an absolute constant, and the notation for two-sided estimates up to an absolute constant.

Without possible confusions, we will write (resp., ) instead of (resp., ).

##### 1.2. Preliminaries: Menagerie of Spaces

Let be a continuous nondecreasing function which vanishes and is continuous at 0. Given a probabilistic space , we define the Orlicz class as the set of all (equivalence classes of) measurable functions on such that for some . We use to define the Morse-Transue space by
and we also introduce the following set:
In general, these three sets are not vector spaces and do not coincide, but we trivially have
We also define the *Luxembourg gauge* on by
This functional is homogeneous and is 0 if and only if a.e., but it is not subadditive a priori.

We say that two functions and as above are equivalent if there exists some constant such that for any large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.

In order to define a good topology on and to get properties convenient for our purpose, we will assume that satisfies the following definition.

*Definition 1.1. *Let . We say that is a growth function of order if it satisfies the two following conditions:(1) is of lower type , that is, for any and at least for large enough;(2) is nonincreasing, at least for every large enough.

We shall say that is a growth function if it is a growth function of order for some .

In particular, a growth function is equivalent to the function which is concave (see [17]). Now, for such a concave growth function of order , while Moreover, if we define (resp., ), then and (that we will simply denote by and ) are two equivalent metrics on for which it is complete. Without loss of generality, we then assume that every growth function that we consider further is concave and diffeomorphic.

*Remark 1.2. *Note that every concave function which vanishes at 0 satisfies the -condition, that is, for some and large enough. This condition is very classical when the function is an Orlicz function, that is, a nondecreasing continuous convex function (see [19, 20]). For large Orlicz class, this condition is natural in order to have vector space.

When we deal with spaces of holomorphic functions, it is very natural to require subharmonicity. Then we will assume that is such that is subharmonic when is holomorphic. We will refer to such a function as a subharmonic-preserving function.

Here are some examples of (concave) growth function that we may consider further.

*Example 1.3. *(1) for and any .

(2) (for ) with large enough, and .

(3) at least for any large, where is an Orlicz function (see Remark 1.2) and is such that is nonincreasing. Note that in this case, is subharmonic whenever is holomorphic, because we have the following.

Proposition 1.4. *Let be as in (3) above. There exists a convex function such that for any large enough.*

* Proof. *Let and be an Orlicz function such that for any . Since is (assumed to be) bijective, we can define the function by for every . Now, using that is convex, is convex since
for any .

It follows that is subharmonic (for holomorphic).

The following lemma gives an upper estimate of the Luxembourg norm of a function in .

Lemma 1.5. *Let be a probabilistic space and let be a growth function of order . For any , one has
*

* Proof. *It is quite identical to that of [10, Lemma 3.9], but we prefer to give the details. Without loss of generality, we may assume that . For every , one has the following, using that is of lower type :
Then the last expression is less than or equal to 1 if and only if .

###### 1.2.1. Large Bergman-Orlicz Spaces

For a subharmonic-preserving growth function and , the weighted Bergman-Orlicz space of the ball consists of those holomorphic functions on which belongs to the Orlicz space . To avoid further confusion, we will denote by the corresponding Luxembourg (quasi)norm and by the quantity . is metric space for the distance or defined by, respectively, and . If , then we recover the usual weighted Bergman space . One checks that we have the followings inclusions: whenever is a growth function of order .

It seems important to us to mention that a linear operator from to with or , where and are two growth functions, is continuous if and only if it maps a bounded set into a bounded set, or equivalently if and only if there exists a constant such that for any such that . If , then is bounded if and only if for as previously.

The next proposition says that the point evaluation functionals are continuous on .

Proposition 1.6. *Let and let be a subharmonic-preserving growth function. For any and any , one has
*

The proof is the same as that of [11, Proposition 1.9] and so is omitted (still use the hypothesis that is subharmonic). We easily deduce from this and the completeness of the following result.

Corollary 1.7. *, endowed with or , is a complete metric space.*

Let and . For , we introduce the following “test’’ function defined by is nothing but the Berezin kernel, hence while . Then, as a consequence of Lemma 1.5, we have whenever is a growth function of order . These functions will be of interest to us later, when proving Carleson theorem for large Bergman-Orlicz spaces.

We now define a maximal operator which was introduced in [11] and that will be bounded on . The definition needs to introduce the sets , defined by and requires the construction of convenient sets based on the following lemma ([11, Lemma 2.1]; we also refer to the forthcoming Section 2.1).

Lemma 1.8. *There exists an integer such that for any , one can find a finite sequence ( depending on ) in with the following properties.*(1)*.*(2)*The sets are mutually disjoint.*(3)*Each point of belongs to at most of the sets .*

From now on, denotes the constant involved in the previous lemma. Let be an integer and let be the corona For any , let be given by Lemma 1.8 putting . For , we set Then we define the sets , for and , by We have both For , we finally define the subset of by . These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function for by where is the characteristic function of . Now we may easily adapt the proof of [11, Proposition 2.2] (which only relies on the subharmonicity of ) to get the following.

Proposition 1.9. *Let be a subharmonic-preserving growth function and let . Then the maximal operator , which carries to , is bounded from to . More precisely, there exists such that for every , one has
**
In particular, a holomorphic function belongs to if and only if belongs to .*

###### 1.2.2. Large Hardy-Orlicz Spaces

Let still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let . The Hardy-Orlicz space of the ball consists of all holomorphic functions on such that where , and where is the Luxembourg norm on the Orlicz space . Note that we can replace by thanks to the subharmonicity of . Because is supposed to be a growth function, we have the following inclusion: for a growth function of order . In particular, every admits a boundary radial limit, denoted by , -almost everywhere on . Let us also note that if is a growth function, then if and only if

Without possible confusion, we will write instead of (resp., instead ). As for Bergman-Orlicz spaces, a linear operator from to some or where and are two growth functions is continuous (bounded) if and only if there exists a constant such that for any such that .

In addition, it is clear that, for any and for any . Therefore, letting tend to −1 in Proposition 1.6, we get the following.

Proposition 1.10. *Let be a subharmonic-preserving growth function. For any and any , one has
*

As a corollary, we have the following.

Corollary 1.11. *Let be a subharmonic-preserving growth function. is a complete metric space (with the equivalent distances induced by and , as usual).*

For and , we introduce the “test’’ function defined for any by It is easily seen that with and that for any . Moreover, let us observe that so that . Therefore, if is a growth function of order , we have, by Lemma 1.5,

It is very convenient to see as a closed subspace of . When is an Orlicz function, this is possible thanks to the representation of any function in by the Poisson integral of its boundary values. This does not work any more in with , even in this case. However, using a radial maximal function, we can still see as a subspace of . We are going to extend this to for a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on : for , It is well known that is a distance on and a pseudodistance on [21, Paragraph 5.1]. It permits to define the Korányi approach region for : Then the maximal function of , associated to Korányi approach region, is given by for any . [17, Theorem 1.3] will be very useful.

Theorem 1.12. *Let be a growth function. Then, for any ,
**
In particular, a holomorphic function belongs to if and only if belongs to .*

From this theorem, we deduce the following one.

Theorem 1.13. *Let be a subharmonic-preserving growth function. Then for every , one has*(1)*;*(2)*;*(3)* is separable. More precisely, the polynomials are dense in .*

* Proof. *Let for . Obviously, , hence (Theorem 1.12). Since is concave and vanishes at 0, we have . Now tends to 0 as goes to 1 for -almost every . By the dominated convergence theorem, (1) follows.

Then and comes from to the subharmonicity of .

We proved in that tends to in for (hence for also). We approach every uniformly on by its Taylor series to get the third assertion.

#### 2. Carleson Embedding Theorems

##### 2.1. Statements of the Results

For and , we define the nonisotropic “ball’’ of by and its analogue in by Let us also denote by the “true’’ balls in . We have and [22].

Let be a positive Borel measure on whose restriction to is absolutely continuous with respect to and let . By definition, is a -Carleson measure if , while it is a vanishing -Carleson measure if when goes to 0. A variant of the well-known Carleson theorem for Hardy spaces [1, 3] ensures that the embedding is bounded (resp., compact) if and only if is a -Carleson measure (resp., a vanishing -Carleson measure).

Similarly, we define the -Bergman Carleson measures (resp., vanishing -Bergman Carleson measures) for weighted Bergman spaces by (resp., ). When , we just speak about -Bergman Carleson measures (resp., vanishing -Bergman Carleson measures). Ueki [7] showed that is bounded (resp., compact) if and only if is a -Bergman Carleson measure (resp., a vanishing -Carleson measure).

In the context of Hardy-Orlicz spaces (resp., weighted Bergman-Orlicz spaces) smaller than (resp., ) (i.e., when the defining function is an *Orlicz function*), much general results were obtained in [9, 10] in the unit disc, and in [11, 12] in the unit ball.

For Hardy-Orlicz (resp., weighted Bergman-Orlicz) spaces larger than (resp., ), we state that the characterizations of the boundedness and compactness of (resp., ), where and are two growth functions such that is nondecreasing at least for large values of (or equivalently nondecreasing, since is increasing and does vanish except in 0), only depend on the growth of at infinity.

Note that if , (with an Orlicz function), or with , then is nondecreasing.

Theorem 2.1. *Let and be two subharmonic-preserving growth functions such that is nondecreasing. Let be a finite positive Borel measure on (whose restriction to is absolutely continuous with respect to ). Then,*(1)* embeds into if and only if there exists some such that, for any ,
uniformly in ,*(2)*the embedding is compact if and only if
uniformly in .**A measure which satisfies (2.4) (resp., (2.5)) will be called a -Carleson measure (resp., a vanishing -Carleson measure).*

For big Bergman-Orlicz spaces, we have the following.

Theorem 2.2. *Let and be two subharmonic-preserving growth functions such that is nondecreasing, and let . Let also be a finite positive Borel measure on . Then,*(1)* embeds into if and only if there exists such that, for any ,
uniformly in ,*(2)*the embedding is compact if and only if
uniformly in .**A measure which satisfies (2.6) (resp., (2.7)) will be called a -Bergman-Carleson measure (resp., a vanishing -Bergman-Carleson measure).*

*Remark 2.3. *By the closed graph theorem, the above embeddings are bounded as soon as they exist.

We immediately deduce from the previous theorems the following corollaries.

Corollary 2.4. *Let and let and , as in (3) of Example 1.3. Let be a finite positive Borel measure on (whose restriction to is absolutely continuous with respect to ) (resp., on ). Then,*(1)* (resp., ) embeds into (resp., ) if and only if is a -Carleson measure (resp., a -Bergman-Carleson measure),*(2)*the embedding (resp., ) is compact if and only if is a vanishing -Carleson measure (resp., a vanishing Bergman-Carleson measure).*

If (or equivalently if and are equivalent), we have the following.

Corollary 2.5. *Let , and let be subharmonic-preserving growth function. Let be a finite positive Borel measure on (whose restriction to is absolutely continuous with respect to ) (resp., on ). Then,*(1)* (resp., ) embeds into (resp., ) if and only if is a Carleson measure (resp., a -Bergman-Carleson measure),*(2)*the embedding (resp., ) is compact if and only if is a vanishing Carleson measure (resp., a vanishing -Bergman-Carleson measure).*

##### 2.2. Proofs of Theorems 2.1 and 2.2

For the compactness parts, we will use a criterion given in [12, Proposition 2.11] and [11, Proposition 2.8]. Its proof is easy to adapt as soon as we have checked that the convergence in (resp., ), for a subharmonic-preserving growth function, implies the convergence on every compact subset of , but it stems from Proposition 1.10 (resp., Proposition 1.6).

Proposition 2.6. *Let , let and be two subharmonic-preserving growth functions and let (resp., ) be a finite positive Borel measure on (resp., ) whose restriction to is absolutely continuous with respect to . One assume that (resp., ) is well defined (hence bounded).*(1)*The two following assertions are equivalent:(a) the canonical embedding (resp., ) is compact;(b)every sequence in the unit ball of (resp., ), which is convergent to 0 uniformly on every compact subset of , is convergent to 0 in (resp., ).*(2)

*If (resp., ), where (resp., ), then the canonical embedding (resp., ) is compact.*

###### 2.2.1. Proof of Theorem 2.1

We assume that the hypothesis of Theorem 2.1 is fulfilled. The proof will be based on two lemmas, whose proofs follow that of Theorem 2.4 and Lemma 2.6 of [12]. These results are refinement of Carleson theorem and are the key to deal with different Hardy-Orlicz spaces which are not classical Hardy spaces. We need to introduce the function , associated to by so that is a Carleson measure if and only if is finite for some .

Theorem 2.7 (see [12, Theorem 2.4]). *There exist two constants and such that, for every continuous on which admits boundary values almost everywhere on , and every finite positive Borel measure on , one has
**
for every and every .*

The next lemma is a the technical key to obtain Carleson theorems in Hardy-Orlicz context. Its proof closely follows that of [12, Lemma 2.6], but we prefer to give the details.

Lemma 2.8. *Let be a finite positive Borel measure on and let and be two subharmonic-preserving growth functions. Let be the constant appearing in Theorem 2.7. Assume that there exist and such that
**
for every . Then, for every such that and every Borel subset of ,
**
where involves a constant which is independent of , , and .*

* Proof. *Let with . With the notations of the statement of the lemma and using Proposition 1.10, the proof of [12, Lemma 2.6] directly yields.
for some constant which depends only on . Now, since and are two concave growth functions, we have
where involve constants which only depend on and . Therefore, (2.12) becomes
Since is nondecreasing and since and and their derivatives do not vanish except in 0, we have, for any ,
It follows that

We now can prove Theorem 2.1.

*Proof of Theorem 2.1. * We assume that is a -Carleson measure, and we intend to show that is well defined (hence bounded). We observe that (2.10) is satisfied for . Indeed we have, for any ,
uniformly in . Now, is nondecreasing (at least for large ) so that we can find such that, for any ,
Therefore, we may and will apply Lemma 2.8 with to get
We finish the proof of using Theorem 1.12 and Inequality (1.7):

For the converse, let , where is the test function introduced at the end of Section 1.2.2, with such that is a growth function of order . According to (1.31), lies in the unit ball of . Let be the (finite) norm of . Using a classical computation which gives that for any , we get for such , hence
for any . We may assume that and since is concave and vanishes at 0, it follows that

(2) We turn to the compactness part. We first prove the sufficient part. Let us assume that is nondecreasing and that is a vanishing -Carleson measure. In particular, is a -Carleson measure and we may apply Proposition 2.6. Then it is sufficient to prove that for every , there exists close enough to 1, such that (where ). We fix and let be in the unit ball of . As in the proof of the sufficient part of , since is nondecreasing, is a vanishing -Carleson measure implying that
for small enough. Then, we apply Lemma 2.8 with and to get the existence of close enough to 1 (independent of ), such that
for any . Now, we may argue as in [12, Lemma 2.13] to prove that, under the assumption that is a -Carleson measure, . Hence
for any , which gives (2).

For the converse, we assume that is compact but that is not a vanishing -Carleson measure, that is there exist , a sequence decreasing to 0 and a sequence such that
Now, let be as in the proof of the necessity in the boundedness part with and such that is growth function of order . Since is bounded in and converges uniformly on every compact subset of , then we must have , because of Proposition 2.6. But, for , we saw that
Since is a growth function of order, let say , we have, for any , . Hence, we get
Therefore, , which is a contradiction.

###### 2.2.2. Proof of Theorem 2.2

We introduce the following function: As for Hardy-Orlicz spaces, we need a refinement of Carleson theorem which allows to deal with weighted Bergman-Orlicz spaces associated to different growth functions. It is [11, Theorem 2.3].

Theorem 2.9. *There exists a constant such that, for every continuous on and every positive finite Borel measure on , one has
**
for every and every .*

Our Carleson theorem for large weighted Bergman-Orlicz spaces is a consequence of the next technical lemma. Its proof is an easy adaptation and combination of the proofs of Lemma 2.8 and [11, Lemma 2.4], so we omit it.

Lemma 2.10. *Let be a finite positive Borel measure on and let and be two subharmonic-preserving growth functions. Assume that there exist , , and such that
**
for every . Then, for every with and every Borel subset of , one has
**
where involves constants which are independent of , , and .*

We now prove Theorem 2.2.

* Proof of Theorem 2.2. *Let and be two subharmonic-preserving growth functions such that is nondecreasing.

(1) We first assume that is a -Bergman Carleson measure, which implies, as in the beginning of the proof of (1) of Theorem 2.1, that condition (2.31) with is fulfilled. Applying (2.31) to , with , gives
We conclude the proof using Proposition 1.9 together with (1.7).

To prove the converse, we argue as in the proof of the corresponding part of Theorem 2.1, using the test function , where has been introduced in Section 1.2.1, with such that is a growth function of order .

(2) The proof of the compactness part of Theorem 2.2 is still similar to that for Hardy-Orlicz spaces: we apply Lemma 2.10 to , with and to show that tends to 0, as tends to 1. Proposition 2.6 then shows that is compact.

For the converse, we procede as for Theorem 2.1 using the test function introduced in the boundedness part above.

#### 3. Applications to Composition Operators

Theorems 2.1 and 2.2 will be used to characterize both boundedness and compactness of composition operators on and . As usual, for holomorphic, will be seen as an embedding operator. To do so, we define the pullback measures and of, respectively, and under : where is the radial limit almost everywhere of and is any Borel subset of . And for any Borel subset of .

By a classical formula for pull-back measures, for any growth function and every polynomial (resp., for any ). Then we extend this equality to the whole space by density of polynomials in (Theorem 1.13) and using Cauchy’s formula.

*Remark 3.1. *Observe that it is not true that for any but it is for any polynomial . Such an equality holds for every function in .

Moreover, we have the following criterion for compactness of composition operators.

Proposition 3.2. *Let and let and be as in Proposition 2.6. Let also be holomorphic. is compact from into (resp., into ) if and only if, for every bounded sequence (resp., ) which converges to 0 on every compact of (resp., ) tends to 0.*

The proof of this proposition is quite similar to that of Proposition 2.6.

Therefore, Theorems 2.1 and 2.2 together with Propositions 2.6 and 3.2 give the following characterizations of the boundedness and compactness of on and .

Theorem 3.3. *Let and be two subharmonic-preserving growth functions such that is nondecreasing. Let also be holomorphic. Then,*(1)* is bounded from into if and only if is a -Carleson measure,*(2)* is compact from into if and only if is a vanishing -Carleson measure.*

For Bergman-Orlicz spaces, we have the following.

Theorem 3.4. *Let and let and be two subharmonic-preserving growth functions such that is nondecreasing. Let also be holomorphic. Then,*(1)* is bounded from into if and only if is a -Bergman Carleson measure,*(2)* is compact from into if and only if is a vanishing -Bergman Carleson measure.*

We immediately deduce the following corollary, which reminds a similar result when the function is an Orlicz function satisfying the -Condition (see [11, 12]).

Corollary 3.5. *Let and let be a subharmonic preserving growth function. Let also be holomorphic. Then,*(1)* is bounded (resp., compact) on if and only if it is bounded (resp., compact) on one (or equivalently every) , if and only if is a Carleson measure (resp., a vanishing Carleson measure),*(2)* is bounded (resp., compact) on if and only if it is bounded (resp., compact) on one (or equivalently every) , if and only if is a Bergman-Carleson measure (resp., a vanishing Bergman-Carleson measure).*

The following corollary shows that the behavior of composition operators between certain classes of different Hardy-Orlicz or Bergman-Orlicz spaces is still the same as that in the classical cases (see [23]).

Corollary 3.6. *Let , let and , and as in (3) of Example 1.3. Let also be holomorphic. Then,*(1)* is bounded (resp., compact) from to if and only if it is bounded (resp., compact) from to , if and only if is a -Carleson measure (resp., a vanishing -Carleson measure),*(2)* is bounded (resp., compact) from to if and only if it is bounded (resp., compact) from to , if and only if is a -Bergman-Carleson measure (resp., a vanishing -Bergman-Carleson measure).*

#### Acknowledgments

The authors would like to thank Ueki for providing his paper [7]. The second author acknowledges support from the Irish Research Council for Science, Engineering and Technology.

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