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Journal of Function Spaces and Applications
Volumeย 2012, Article IDย 792763, 21 pages
http://dx.doi.org/10.1155/2012/792763
Research Article

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

1Dรฉpartement de Mathรฉmatiques, Universitรฉ Paris-Sud, Bรขtiment 425, 91405 Orsay, France
2School of Mathematics, Trinity College, Dublin 2, Ireland

Received 24 October 2011; Accepted 20 February 2012

Academic Editor: Miroslavย Englis

Copyright ยฉ 2012 Stรฉphane Charpentier and Benoรฎt 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 ๐œ‡ for which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) space ๐ปฮจ1 (resp., ๐ดฮจ1๐›ผ) of the unit ball of โ„‚๐‘ embeds boundedly or compactly into the Orlicz space ๐ฟฮจ2(๐”น๐‘,๐œ‡) (resp., ๐ฟฮจ2(๐”น๐‘,๐œ‡)), when the defining functions ฮจ1 and ฮจ2 are growth functions such that ๐ฟ1โŠ‚๐ฟฮจ๐‘— for ๐‘—โˆˆ{1,2}, and such that ฮจ2/ฮจ1 is nondecreasing. We apply our result to the characterization of the boundedness and compactness of composition operators from ๐ปฮจ1 (resp., ๐ดฮจ1๐›ผ) into ๐ปฮจ2 (resp., ๐ดฮจ2๐›ผ).

1. Introduction and Preliminaries

1.1. Introduction

Let ๐”น๐‘={๐‘ง=(๐‘ง1,โ€ฆ,๐‘ง๐‘)โˆˆโ„‚๐‘โˆถ|๐‘ง|2=โˆ‘๐‘๐‘–=1|๐‘ง๐‘–|2<1} and ๐•Š๐‘=๐œ•๐”น๐‘, ๐‘โ‰ฅ1, denoting, respectively, the unit ball and the unit sphere of โ„‚๐‘. For ๐‘=1, we denote by ๐”ป the unit disc of the complex plane.

For a large class of spaces ๐‘‹ of holomorphic functions in the unit disc or the unit ball, characterizations of the boundedness and compactness of the canonical embedding ๐‘‹โ†ช๐ฟ๐‘(๐œ‡) have been given and applied to different areas, for example, interpolation, multipliers, integral operators, composition operators, and so forth. These results are known as Carlesonโ€™s type theorems.

First, when ๐‘‹=๐ป๐‘(๐”ป), Carleson [1] proved that ๐ป๐‘(๐”ป)โ†ช๐ฟ๐‘(๐œ‡) if and only if the finite positive Borel measure ๐œ‡ on ๐”ป (or ๐”ป) is a so-called Carleson measure. This result was extended to the unit ball by Hรถrmander [2], whose proof was simplified by Power [3]. Duren [4] characterized those measures ๐œ‡ such that ๐ป๐‘(๐”ป)โ†ช๐ฟ๐‘ž(๐œ‡), with 0<๐‘โ‰ค๐‘ž<โˆž, in terms of (๐‘ž/๐‘)-Carleson measures. For the unweighted and weighted Bergman spaces ๐ด๐‘๐›ผ(๐”น๐‘), ๐‘โ‰ฅ1, similar results were obtained by Cima and Wogen [5], Luecking [6], Ueki [7]. We recall also that the compactness of ๐‘‹โ†ช๐ฟ๐‘(๐œ‡) was also characterized in the previous cases, in terms of vanishing Carlesonโ€™s type measures. It is usual to assume that when the measure ๐œ‡ is defined on ๐”น๐‘, then ๐œ‡โˆฃ๐•Š๐‘ is absolutely continuous with respect to ๐•Š๐‘. 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 ๐ป๐‘(๐”น๐‘)โ†ช๐ฟ๐‘ž(๐œ‡) are not always satisfied and just depend on the ratio ๐‘ž/๐‘ for 0<๐‘โ‰ค๐‘ž<โˆž; in particular, when ๐‘=๐‘ž, they are independent of ๐‘. Now, since the restriction to ๐•Š๐‘ of the finite positive Borel ๐œ‡ is assumed to be absolutely continuous with respect to the Lebesgue measure, then it is trivial that ๐ปโˆžโ†ช๐ฟโˆž(๐œ‡) always holds. On the contrary, the compactness of this inclusion implies strong condition on ๐œ‡. This suggests to think about this question when the space ๐‘‹ is intercalated between every ๐ป๐‘ and ๐ปโˆž.

This motivation is reinforced by a second observation: if the measure ๐œ‡๐œ™ is the pull-back measure of the invariant-rotation measure on ๐•Š๐‘ under a holomorphic map ๐œ™โˆถ๐”นNโ†’๐”น๐‘, ๐‘โ‰ฅ1, then ๐œ‡๐œ™ is always a Carleson measure when ๐‘=1 (this is the Littlewood Subordination Principle, see [8]), but it is not systematic for ๐‘>1. For the compactness, there is still a big gap between ๐ป๐‘ and ๐ปโˆž for any ๐‘โ‰ฅ1. This observation is directly connected to the study of composition operators ๐ถ๐œ™ on ๐ป๐‘, which are defined by ๐ถ๐œ™(๐‘“)=๐‘“โˆ˜๐œ™ for ๐‘“โˆˆ๐ป(๐”น๐‘), and which may be seen as the embedding operators ๐ป๐‘โ†ช๐ฟ๐‘(๐œ‡๐œ™). By the way, this leads the authors of [9, 10] to state Carleson theorems for Hardy-Orlicz and Bergman-Orlicz spaces (resp., denoted by ๐ปฮจ(๐”ป) and ๐ดฮจ(๐”ป)) in the unit disc, when the defining function ฮจ is an Orlicz function. These spaces appear as good candidates for generalizing ๐ป๐‘ and ๐ด๐‘ spaces, 1โ‰ค๐‘<โˆž, and for covering the gap with ๐ปโˆž. 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 ฮจ. Indeed, he showed that, when ๐œ‡=๐œ‡๐œ™ is the pull-back measure under a holomorphic self-map ๐œ™ of ๐”น๐‘, then ๐ปฮจ always embeds into ๐ฟฮจ(๐œ‡) whenever ฮจ satisfied a fast growth condition (namely, the ฮ”2-Condition, that implies ฮจ(๐‘ฅ)โ‰ฅ๐‘’๐‘ฅ for large value of ๐‘ฅ and means that we are close to ๐ปโˆž). Similarly, it was shown in [13] that if ๐ปฮจ compactly embeds into ๐ฟฮจ(๐œ‡) for every ฮจ, then ๐ปโˆžโ†ช๐ฟโˆž(๐œ‡) compactly (the converse easily holds). We mention that there is no ฮจ such that ๐ปฮจโ†ช๐ฟฮจ(๐œ‡) if and only if ๐ปโˆžโ†ช๐ฟโˆž(๐œ‡) [14], and that such previous results are not true for arbitrary measure ๐œ‡. Yet, a link has been made between the involved Carleson conditions and the type of growth of the Orlicz function ฮจ. This is also strengthened by the fact that if the Orlicz function ฮจ is dominated by a power function (exactly ฮจ satisfies the ฮ”2-Condition), then ๐ปฮจโ†ช๐ฟฮจ(๐œ‡) boundedly (resp., compactly) if and only if ๐ป๐‘โ†ช๐ฟ๐‘(๐œ‡), that is, if and only if ๐œ‡ 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 ๐ป1(๐”น๐‘). It seems that nothing has been done in this direction, except for explicit functions ฮจ. In particular, the second author gave a necessary and sufficient condition for the inclusion ๐ปฮจ(๐”น๐‘)โ†ช๐ฟ๐‘ž(๐œ‡) to be bounded, when ฮจ(๐‘ก)=(๐‘ก/log(๐‘’+๐‘ก))๐‘ , 0<๐‘ โ‰ค1, and 1โ‰ค๐‘ž<โˆž [15]. Moreover, [16] characterized Carleson measures for ๐ดฮจ๐›ผ(๐”ป), where ฮจ=log๐‘+, 1โ‰ค๐‘<โˆž, that is, when ๐ดฮจ๐›ผ(๐”ป) is the area Nevanlinna space. These measures reveal to be those ๐œ‡ for which ๐ด๐‘๐›ผโ†ช๐ฟ๐‘(๐œ‡) 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 ฮจ, 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 ๐œ‡ such that ๐ปฮจ1(๐”น๐‘)โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) (resp., ๐ดฮจ1๐›ผ(๐”น๐‘)โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡)) is bounded or compact, when ฮจ1 and ฮจ2 are two growth functions (i.e., for which ๐ป1โŠ‚๐ปฮจ๐‘–โŠ‚๐ป๐‘ (resp., ๐ด1๐›ผโŠ‚๐ดฮจ๐‘–๐›ผโŠ‚๐ด๐‘๐›ผ) for some ๐‘, ๐‘–,๐‘—โˆˆ{1,2}), and such that ฮจ2โ€‰โ€‰grows faster than ฮจ1. It appears that if ฮจ1=ฮจ2, then such measures are exactly those which are Carleson (resp., Bergman-Carleson) measures for some 0<๐‘<1. For the Bergman-Orlicz case, these results let one think that, between the area Nevanlinna class [16] and ๐ด1๐›ผ, there is no difference regarding to Carleson theorems, whenever the defining functions ฮจ share some natural properties. This is in contrast with what happens between ๐ด1๐›ผ and ๐ปโˆž.

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 ๐‘ง,๐‘คโˆˆโ„‚๐‘, the euclidean inner product of ๐‘ง and ๐‘ค will be denoted by โŸจ๐‘ง,๐‘คโŸฉ, that is, โˆ‘โŸจ๐‘ง,๐‘คโŸฉ=๐‘๐‘–=1๐‘ง๐‘–๐‘ค๐‘–; the notation |โ‹…| will stand for the associated norm, as well as for the modulus of a complex number.

๐œŽ will stand for the invariant-rotation measure on the unit sphere. For ๐›ผ>โˆ’1,๐‘ฃ๐›ผ will be the measure on ๐”น๐‘ defined by ๐‘‘๐‘ฃ๐›ผ=๐‘๐›ผ(1โˆ’|๐‘ง|)๐›ผ๐‘‘๐‘ฃ, where ๐‘ฃ is the Lebesgue measure on ๐”น๐‘ and ๐‘๐›ผ is the constant of normalization.

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

Without possible confusions, we will write ๐ป๐œ“ (resp., ๐ด๐œ“๐›ผ) instead of ๐ป๐œ“(๐”น๐‘) (resp., ๐ด๐œ“๐›ผ(๐”น๐‘)).

1.2. Preliminaries: Menagerie of Spaces

Let ฮจโˆถ[0,โˆž)โ†’[0,โˆž) be a continuous nondecreasing function which vanishes and is continuous at 0. Given a probabilistic space (ฮฉ,โ„™), we define the Orlicz class ๐ฟฮจ(ฮฉ,โ„™) as the set of all (equivalence classes of) measurable functions ๐‘“ on ฮฉ such that โˆซฮฉฮจ(|๐‘“|/๐ถ)๐‘‘โ„™<โˆž for some 0<๐ถ<โˆž. We use to define the Morse-Transue space ๐‘€ฮจ(ฮฉ,โ„™) by๐‘€ฮจ๎‚ป๎€œ(ฮฉ,โ„™)=๐‘“โˆถฮฉโŸถโ„‚measurable;ฮฉฮจ๎‚ต||๐‘“||๐ถ๎‚ถ๎‚ผ,<โˆžforany๐ถ>0(1.1) and we also introduce the following set:โ„’ฮจ๎‚ป๎€œ(ฮฉ,โ„™)=๐‘“โˆถฮฉโŸถโ„‚measurable;ฮฉฮจ๎€ท||๐‘“||๎€ธ๎‚ผ<โˆž.(1.2) In general, these three sets are not vector spaces and do not coincide, but we trivially have๐‘€ฮจ(ฮฉ,โ„™)โŠ‚โ„’ฮจ(ฮฉ,โ„™)โŠ‚๐ฟฮจ(ฮฉ,โ„™).(1.3) We also define the Luxembourg gauge on ๐ฟฮจ(ฮฉ,โ„™) byโ€–๐‘“โ€–ฮจ๎‚ต๎€œ=inf๐œ†>0,ฮฉฮจ๎‚ต||๐‘“||๐œ†๎‚ถ๎‚ถ.โ‰ค1(1.4) This functional is homogeneous and is 0 if and only if ๐‘“=0โ„™โˆ’a.e., but it is not subadditive a priori.

We say that two functions ฮจ1 and ฮจ2 as above are equivalent if there exists some constant ๐‘ such that๐‘ฮจ1(๐‘๐‘ฅ)โ‰คฮจ2(๐‘ฅ)โ‰ค๐‘โˆ’1ฮจ1๎€ท๐‘โˆ’1๐‘ฅ๎€ธ,(1.5) for any ๐‘ฅ large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.

In order to define a good topology on ๐ฟฮจ(ฮฉ,โ„™) and to get properties convenient for our purpose, we will assume that ฮจ satisfies the following definition.

Definition 1.1. Let 0<๐‘โ‰ค1. We say that ฮจโˆถ[0,โˆž)โ†’[0,โˆž) is a growth function of order ๐‘ if it satisfies the two following conditions:(1)ฮจ is of lower type ๐‘, that is, ฮจ(๐‘ฆ๐‘ฅ)โ‰ค๐‘ฆ๐‘ฮจ(๐‘ฅ) for any 0<๐‘ฆโ‰ค1 and at least for ๐‘ฅ large enough;(2)๐‘ฅโ†ฆฮจ(๐‘ฅ)/๐‘ฅ is nonincreasing, at least for every ๐‘ฅ large enough.

We shall say that ๐œ“ is a growth function if it is a growth function of order ๐‘ for some 0<๐‘โ‰ค1.

In particular, a growth function ฮจ is equivalent to the function โˆซ๐‘ฅโ†ฆ๐‘ฅ0(ฮจ(๐‘ )/๐‘ )๐‘‘๐‘  which is concave (see [17]). Now, for such a concave growth function of order ๐‘,๐ฟ1(ฮฉ,โ„™)โŠ‚๐‘€ฮจ(ฮฉ,โ„™)=โ„’ฮจ(ฮฉ,โ„™)=๐ฟฮจ(ฮฉ,โ„™)โŠ‚๐ฟ๐‘(ฮฉ,โ„™),(1.6)โ€–๐‘“โ€–ฮจ๎‚ป๎€œโ‰ฒminฮฉฮจ๎€ท||๐‘“||๎€ธ๎‚ต๎€œ๐‘‘โ„™,ฮฉฮจ๎€ท||๐‘“||๎€ธ๎‚ถ๐‘‘โ„™๐‘๎‚ผ,(1.7) whileโ€–๐‘“โ€–ฮจ๎€œโˆถ=ฮฉฮจ๎€ท||๐‘“||๎€ธ๎€ฝ๐‘‘โ„™โ‰ฒmaxโ€–๐‘“โ€–ฮจ,โ€–๐‘“โ€–๐‘ฮจ๎€พ.(1.8) Moreover, if we define ๐‘‘ฮจ(๐‘“,๐‘”)=โ€–๐‘“โˆ’๐‘”โ€–ฮจ (resp., ๐‘‘ฮจ(๐‘“,๐‘”)=โ€–๐‘“โˆ’๐‘”โ€–ฮจ), then ๐‘‘ฮจ and ๐‘‘ฮจ (that we will simply denote by โ€–โ‹…โ€–ฮจ and โ€–โ‹…โ€–ฮจ) are two equivalent metrics on ๐ฟฮจ(ฮฉ,โ„™) 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 ฮ”2-condition, that is, ฮจ(2๐‘ฅ)โ‰ค๐พฮจ(๐‘ฅ) for some ๐พ>1 and ๐‘ฅ large enough. This condition is very classical when the function ฮจ 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 ฮจ is such that ฮจ(|๐‘“|) is subharmonic when ๐‘“ 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)โ€‰โ€‰ฮจ1(๐‘ฅ)=๐‘ฅ๐‘ for 0<๐‘โ‰ค1 and any ๐‘ฅโ‰ฅ๐‘ฅ0.
(2)โ€‰โ€‰ฮจ2(๐‘ฅ)=๐‘ฅ๐‘log๐‘ž(๐ถ+๐‘ฅ) (for ๐‘ฅโ‰ฅ๐‘ฅ0) with ๐ถ>0 large enough, 0<๐‘โ‰ค1 and ๐‘žโ‰ฅ1.
(3)โ€‰ ฮจ3(๐‘ฅ)=ฮฆ๐‘(๐‘ฅ) at least for any ๐‘ฅ large, where ฮฆ is an Orlicz function (see Remark 1.2) and ๐‘>0 is such that ฮฆ๐‘(๐‘ฅ)/๐‘ฅ is nonincreasing. Note that in this case, ฮจ(|๐‘“|) is subharmonic whenever ๐‘“ is holomorphic, because we have the following.

Proposition 1.4. Let ฮจ be as in (3) above. There exists a convex function ๐œ“ such that ฮจ(๐‘ฅ)=๐œ“(๐‘ฅ๐‘) for any ๐‘ฅ>0 large enough.

Proof. Let ๐‘>0 and ฮฆ be an Orlicz function such that ฮจ(๐‘ฅ)=ฮฆ๐‘(๐‘ฅ) for any ๐‘ฅโ‰ฅ0. Since ฮจ is (assumed to be) bijective, we can define the function ๐œ“ by ๐œ“(๐‘ฅ๐‘)=ฮฆ๐‘(๐‘ฅ) for every ๐‘ฅ. Now, using that ฮฆ is convex, ๐œ“ is convex since ๐œ“(๐‘ฅ)๐‘ฅ=๎ƒฉฮฆ๎€ท๐‘ฅ1/๐‘๎€ธ๐‘ฅ1/๐‘๎ƒช๐‘โ‰ค๎ƒฉฮฆ๎€ท๐‘ฆ1/๐‘๎€ธ๐‘ฆ1/๐‘๎ƒช๐‘=๐œ“(๐‘ฆ)๐‘ฆ,(1.9) for any 0โ‰ค๐‘ฅโ‰ค๐‘ฆ.

It follows that ฮจ(|๐‘“|)=๐œ“(|๐‘“|๐‘) is subharmonic (for ๐‘“ holomorphic).

The following lemma gives an upper estimate of the Luxembourg norm of a function in ๐ฟโˆž(ฮฉ,โ„™).

Lemma 1.5. Let (ฮฉ,โ„™) be a probabilistic space and let ฮจ be a growth function of order ๐‘. For any ๐‘“โˆˆ๐ฟโˆž(ฮฉ,โ„™), one has โ€–๐‘“โ€–ฮจโ‰คโ€–๐‘“โ€–โˆžฮจโˆ’1๎€ทโ€–๐‘“โ€–โˆž/โ€–๐‘“โ€–๐‘๐‘๎€ธ.(1.10)

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 โ€–๐‘“โ€–โˆž=1. For every ๐ถ>0, one has the following, using that ฮจ is of lower type ๐‘: ๎€œฮฉฮจ๎‚ต||๐‘“||๐ถ๎‚ถ๎€œ๐‘‘โ„™โ‰คฮฉ||๐‘“||๐‘ฮจ๎‚€1๐ถ๎‚๐‘‘โ„™โ‰คโ€–๐‘“โ€–๐‘๐‘ฮจ๎‚€1๐ถ๎‚.(1.11) Then the last expression is less than or equal to 1 if and only if ๐ถโ‰ฅ1/ฮจโˆ’1(1/โ€–๐‘“โ€–๐‘๐‘).

1.2.1. Large Bergman-Orlicz Spaces

For ฮจ a subharmonic-preserving growth function and ๐›ผ>โˆ’1, the weighted Bergman-Orlicz space ๐ดฮจ๐›ผ of the ball consists of those holomorphic functions on ๐”น๐‘ which belongs to the Orlicz space ๐ฟฮจ(๐”น๐‘,๐‘ฃ๐›ผ). To avoid further confusion, we will denote by โ€–โ‹…โ€–๐›ผ,ฮจ the corresponding Luxembourg (quasi)norm and by โ€–โ‹…โ€–ฮจ๐›ผ the quantity โˆซ๐”น๐‘ฮจ(|๐‘“|)๐‘‘๐‘ฃ๐›ผ. ๐ดฮจ๐›ผ is metric space for the distance ๐‘‘๐›ผ,ฮจ or ๐‘‘ฮจ๐›ผ defined by, respectively, ๐‘‘๐›ผ,ฮจ(๐‘“,๐‘”)=โ€–๐‘“โˆ’๐‘”โ€–๐›ผ,ฮจ and ๐‘‘ฮจ๐›ผ(๐‘“,๐‘”)=โ€–๐‘“โˆ’๐‘”โ€–ฮจ๐›ผ. If ฮจ(๐‘ก)=๐‘ก๐‘, then we recover the usual weighted Bergman space ๐ด๐‘๐›ผ. One checks that we have the followings inclusions:๐ด1๐›ผโŠ‚๐ดฮจ๐›ผโŠ‚๐ด๐‘๐›ผ,(1.12) whenever ฮจ is a growth function of order ๐‘.

It seems important to us to mention that a linear operator ๐‘‡ from ๐ดฮจ1๐›ผ to ๐‘‹ with ๐‘‹=๐ฟฮจ2(๐”น๐‘,๐‘ฃ๐›ผ) or ๐‘‹=๐ดฮจ2๐›ผ, where ฮจ1 and ฮจ2 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 ๐ถ>0 such that๎‚€โ€–max๐‘‡(๐‘“)โ€–๐›ผ,ฮจ2,โ€–๐‘‡(๐‘“)โ€–ฮจ2๐›ผ๎‚โ‰ค๐ถ,(1.13) for any ๐‘“โˆˆ๐ดฮจ1๐›ผ such that min(โ€–๐‘“โ€–๐›ผ,ฮจ1,โ€–๐‘“โ€–ฮจ1๐›ผ)โ‰ค1. If ๐‘‹=โ„‚, then ๐‘‡ is bounded if and only if |๐‘‡(๐‘“)|โ‰ค๐ถ for ๐‘“ as previously.

The next proposition says that the point evaluation functionals are continuous on ๐ดฮจ๐›ผ.

Proposition 1.6. Let ๐›ผ>โˆ’1 and let ฮจ be a subharmonic-preserving growth function. For any ๐‘Žโˆˆ๐”น๐‘ and any ๐‘“โˆˆ๐ดฮจ๐›ผ, one has ||||๐‘“(๐‘Ž)โ‰คฮจโˆ’1๎ƒฉ๎‚ต2๎‚ถ1โˆ’|๐‘Ž|(๐‘+๐›ผ+1)๎ƒชโ€–๐‘“โ€–๐›ผ,ฮจ.(1.14)

The proof is the same as that of [11, Proposition 1.9] and so is omitted (still use the hypothesis that ฮจ(|๐‘“|) is subharmonic). We easily deduce from this and the completeness of ๐ฟฮจ the following result.

Corollary 1.7. ๐ดฮจ๐›ผ, endowed with ๐‘‘๐›ผ,ฮจ or ๐‘‘ฮจ๐›ผ, is a complete metric space.

Let ๐‘>0 and ๐›ผ>โˆ’1. For ๐‘Žโˆˆ๐”น๐‘, we introduce the following โ€œtestโ€™โ€™ function ๐‘“๐‘Ž,๐›ผ,๐‘ defined by๐‘“๐‘Ž,๐›ผ,๐‘=๎ƒฉ1โˆ’|๐‘Ž|2(1โˆ’โŸจ๐‘ง,๐‘ŽโŸฉ)2๎ƒช(๐‘+๐›ผ+1)/๐‘.(1.15)|๐‘“๐‘Ž,๐›ผ,๐‘|๐‘ is nothing but the Berezin kernel, hence โ€–๐‘“๐‘Ž,๐›ผ,๐‘โ€–โˆž=((1+|๐‘Ž|)/(1โˆ’|๐‘Ž|))(๐‘+๐›ผ+1)/๐‘ while โ€–๐‘“๐‘Ž,๐›ผ,๐‘โ€–๐‘=1. Then, as a consequence of Lemma 1.5, we haveโ€–โ€–๐‘“๐‘Ž,๐›ผ,๐‘โ€–โ€–๐›ผ,ฮจโ‰ค๎‚ต2๎‚ถ1โˆ’|๐‘Ž|(๐‘+๐›ผ+1)/๐‘1ฮจโˆ’1๎€ท(2/1โˆ’|๐‘Ž|)(๐‘+๐›ผ+1)๎€ธ,(1.16) whenever ฮจ is a growth function of order ๐‘. 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 ๐ดฮจ๐›ผ. The definition needs to introduce the sets ๐‘„(๐œ,โ„Ž), defined by๐‘„๎€ฝ(๐œ,โ„Ž)=๐‘งโˆˆ๐•Š๐‘,||||๎€พ1โˆ’โŸจ๐‘ง,๐œโŸฉ<โ„Ž,(1.17) 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 ๐‘€>0 such that for any 0<๐‘Ÿ<1, one can find a finite sequence {๐œ‰๐‘˜}๐‘š๐‘˜=1 (๐‘š depending on ๐‘Ÿ) in ๐•Š๐‘ with the following properties.(1)๐•Š๐‘=โ‹ƒ๐‘˜๐‘„(๐œ‰๐‘˜,๐‘Ÿ).(2)The sets ๐‘„(๐œ‰๐‘˜,๐‘Ÿ/4) are mutually disjoint.(3)Each point of ๐•Š๐‘ belongs to at most ๐‘€ of the sets ๐‘„(๐œ‰๐‘˜,4๐‘Ÿ).

From now on, ๐‘€ denotes the constant involved in the previous lemma. Let ๐‘›โ‰ฅ0 be an integer and let ๐ถ๐‘› be the corona๐ถ๐‘›=๎‚ป๐‘งโˆˆ๐”น๐‘1,1โˆ’2๐‘›1โ‰ค|๐‘ง|<1โˆ’2๐‘›+1๎‚ผ.(1.18) For any ๐‘›โ‰ฅ0, let (๐œ‰๐‘›,๐‘˜)๐‘˜โŠ‚๐•Š๐‘ be given by Lemma 1.8 putting ๐‘Ÿ=1/2๐‘›. For ๐‘˜โ‰ฅ0, we set๐‘‡0,๐‘˜=๎‚ป๐‘งโˆˆ๐”น๐‘๐‘งโงต{0},๎€ท๐œ‰|๐‘ง|โˆˆ๐‘„0,๐‘˜๎€ธ๎‚ผ,1โˆช{0}.(1.19) Then we define the sets ๐‘‡๐‘›,๐‘˜, for ๐‘›โ‰ฅ1 and ๐‘˜โ‰ฅ0, by๐‘‡๐‘›,๐‘˜=๎‚ป๐‘งโˆˆ๐”น๐‘๐‘งโงต{0},๎‚€๐œ‰|๐‘ง|โˆˆ๐‘„๐‘›,๐‘˜,12๐‘›๎‚๎‚ผ.(1.20) We have both๎š๐‘›โ‰ฅ0๐ถ๐‘›=๐”น๐‘,๎š๐‘˜โ‰ฅ0๐‘‡0,๐‘˜=๐”น๐‘,๎š๐‘˜โ‰ฅ0๐‘‡๐‘›,๐‘˜=๐”น๐‘โงต{0},๐‘›โ‰ฅ1.(1.21) For (๐‘›,๐‘˜)โˆˆโ„•2, we finally define the subset ฮ”(๐‘›,๐‘˜) of ๐”น๐‘ by ฮ”(๐‘›,๐‘˜)=๐ถ๐‘›โˆฉ๐‘‡๐‘›,๐‘˜. These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function ฮ›๐‘“ for ๐‘“โˆˆ๐ดฮจ๐›ผ(๐”น๐‘) byฮ›๐‘“=๎“๐‘›,๐‘˜โ‰ฅ0supฮ”(๐‘›,๐‘˜)๎€ท||||๎€ธ๐œ’๐‘“(๐‘ง)ฮ”(๐‘›,๐‘˜),(1.22) where ๐œ’ฮ”(๐‘›,๐‘˜) is the characteristic function of ฮ”(๐‘›,๐‘˜). Now we may easily adapt the proof of [11, Proposition 2.2] (which only relies on the subharmonicity of ฮจ(|๐‘“|)) to get the following.

Proposition 1.9. Let ฮจ be a subharmonic-preserving growth function and let ๐›ผ>โˆ’1. Then the maximal operator ฮ›, which carries ๐‘“ to ฮ›๐‘“, is bounded from ๐ดฮจ๐›ผ to ๐ฟฮจ(๐”น๐‘,๐‘ฃ๐›ผ). More precisely, there exists ๐ตโ‰ฅ1 such that for every ๐‘“โˆˆ๐ดฮจ๐›ผ, one has โ€–โ€–ฮ›๐‘“โ€–โ€–๐ฟฮจ(๐”น๐‘,๐‘ฃ๐›ผ)โ‰ค๐ตโ€–๐‘“โ€–๐›ผ,ฮจ.(1.23) In particular, a holomorphic function ๐‘“ belongs to ๐ดฮจ๐›ผ if and only if ฮ›๐‘“ belongs to ๐ฟฮจ(๐”น๐‘,๐‘ฃ๐›ผ).

1.2.2. Large Hardy-Orlicz Spaces

Let ฮจ still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let (ฮฉ,โ„™)=(๐•Š๐‘,๐œŽ). The Hardy-Orlicz space ๐ปฮจ of the ball consists of all holomorphic functions ๐‘“ on ๐”น๐‘ such thatโ€–๐‘“โ€–๐ปฮจโˆถ=sup0<๐‘Ÿ<1โ€–โ€–๐‘“๐‘Ÿโ€–โ€–ฮจ<โˆž,(1.24) where ๐‘“๐‘Ÿ(๐‘ง)=๐‘“(๐‘Ÿ๐‘ง), and where โ€–โ‹…โ€–ฮจ is the Luxembourg norm on the Orlicz space ๐ฟฮจ(๐•Š๐‘,๐œŽ). Note that we can replace sup0<๐‘Ÿ<1โ€–๐‘“๐‘Ÿโ€–ฮจ<โˆž by lim๐‘Ÿโ†’1โ€–๐‘“๐‘Ÿโ€–ฮจ<โˆž thanks to the subharmonicity of ฮจ(|๐‘“|). Because ฮจ is supposed to be a growth function, we have the following inclusion:๐ป1โŠ‚๐ปฮจโŠ‚๐ป๐‘(1.25) for a growth function ฮจ of order ๐‘. In particular, every ๐‘“โˆˆ๐ปฮจ admits a boundary radial limit, denoted by ๐‘“โˆ—, ๐œŽ-almost everywhere on ๐•Š๐‘. Let us also note that if ฮจ is a growth function, then ๐‘“โˆˆ๐ปฮจ if and only ifโ€–๐‘“โ€–ฮจ๐ปฮจโˆถ=sup0<๐‘Ÿ<1โ€–โ€–๐‘“๐‘Ÿโ€–โ€–ฮจ=lim๐‘Ÿโ†’1โ€–โ€–๐‘“๐‘Ÿโ€–โ€–ฮจ<โˆž(Inequalities(1.7)and(1.8)).(1.26)

Without possible confusion, we will write โ€–โ‹…โ€–ฮจ instead of โ€–โ‹…โ€–๐ปฮจ (resp., โ€–โ‹…โ€–ฮจ instead โ€–โ‹…โ€–ฮจ๐ปฮจ). As for Bergman-Orlicz spaces, a linear operator ๐‘‡ from ๐ปฮจ1 to some ๐ฟฮจ2(๐•Š๐‘,๐œŽ) or ๐ปฮจ2 where ฮจ1 and ฮจ2 are two growth functions is continuous (bounded) if and only if there exists a constant ๐ถ>0 such that๎€ทmaxโ€–๐‘‡(๐‘“)โ€–ฮจ2,โ€–๐‘‡(๐‘“)โ€–ฮจ2๎€ธโ‰ค๐ถ(1.27) for any ๐‘“โˆˆ๐ปฮจ1 such that min(โ€–๐‘“โ€–ฮจ1,โ€–๐‘“โ€–ฮจ1)โ‰ค1.

In addition, it is clear that, for any ๐›ผ>โˆ’1,๐ดฮจ๐›ผโŠƒ๐ปฮจ and โ€–๐‘“โ€–๐›ผ,ฮจโ‰คโ€–๐‘“โ€–ฮจ for any ๐‘“โˆˆ๐ปฮจ. Therefore, letting ๐›ผ tend to โˆ’1 in Proposition 1.6, we get the following.

Proposition 1.10. Let ฮจ be a subharmonic-preserving growth function. For any ๐‘Žโˆˆ๐”น๐‘ and any ๐‘“โˆˆ๐ปฮจ, one has ||||๐‘“(๐‘Ž)โ‰คฮจโˆ’1๎ƒฉ๎‚ต1๎‚ถ1โˆ’|๐‘Ž|๐‘๎ƒชโ€–๐‘“โ€–ฮจ.(1.28)

As a corollary, we have the following.

Corollary 1.11. Let ฮจ be a subharmonic-preserving growth function. ๐ปฮจ is a complete metric space (with the equivalent distances induced by โ€–โ‹…โ€–ฮจ and โ€–โ‹…โ€–ฮจ, as usual).

For ๐‘Žโˆˆ๐”น๐‘ and ๐‘>0, we introduce the โ€œtestโ€™โ€™ function ๐‘“๐‘Ž,๐‘ defined for any ๐‘งโˆˆ๐”น๐‘ by๐‘“๐‘Ž,๐‘๎‚ต(๐‘ง)=1โˆ’|๐‘Ž|๎‚ถ1โˆ’โŸจ๐‘ง,๐‘ŽโŸฉ2๐‘/๐‘.(1.29) It is easily seen that ๐‘“๐‘Ž,๐‘โˆˆ๐ปโˆž with โ€–๐‘“๐‘Ž,๐‘โ€–โˆž=1 and that |๐‘“๐‘Ž,๐‘(๐‘ง)|โ‰ค1 for any ๐‘งโˆˆ๐•Š๐‘. Moreover, let us observe that||๐‘“๐‘Ž,๐‘||(๐‘ง)๐‘=๎‚ต1โˆ’|๐‘Ž|๎‚ถ1+|๐‘Ž|๐‘๐‘ƒ๎‚ต๐‘Ž|๐‘Ž|๐‘ง,๎‚ถ,|๐‘Ž|(1.30) so that โ€–๐‘“๐‘Ž,๐‘โ€–๐‘=((1โˆ’|๐‘Ž|)/(1+|๐‘Ž|))๐‘/๐‘. Therefore, if ฮจ is a growth function of order ๐‘, we have, by Lemma 1.5,โ€–โ€–๐‘“๐‘Ž,๐‘โ€–โ€–ฮจโ‰ค1ฮจโˆ’1๎‚€โ€–โ€–๐‘“1/๐‘Ž,๐‘โ€–โ€–๐‘๎‚โ‰ค1ฮจโˆ’1๎€ท(1/(1โˆ’|๐‘Ž|))๐‘/๐‘๎€ธ.(1.31)

It is very convenient to see ๐ปฮจ as a closed subspace of ๐ฟฮจ(๐•Š๐‘). When ฮจ is an Orlicz function, this is possible thanks to the representation of any function in ๐ป1 by the Poisson integral of its boundary values. This does not work any more in ๐ป๐‘ with 0<๐‘<1, even in this case. However, using a radial maximal function, we can still see ๐ป๐‘ as a subspace of ๐ฟ๐‘. We are going to extend this to ๐ปฮจ for ฮจ a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on ๐”น๐‘: for (๐‘ง,๐‘ค)โˆˆ๐”น๐‘,๎”๐‘‘(๐‘ง,๐‘ค)=||||.1โˆ’โŸจ๐‘ง,๐‘คโŸฉ(1.32) It is well known that ๐‘‘ is a distance on ๐•Š๐‘ and a pseudodistance on ๐”น๐‘ [21, Paragraph 5.1]. It permits to define the Korรกnyi approach region ฮ“(๐œ) for ๐œโˆˆ๐•Š๐‘:๎€ฝฮ“(๐œ)=๐‘งโˆˆ๐”น๐‘,๐‘‘(๐‘ง,๐œ)2<1โˆ’|๐‘ง|2๎€พ.(1.33) Then the maximal function ๐‘๐‘“ of ๐‘“, associated to Korรกnyi approach region, is given by๐‘๐‘“(๐œ)=sup๐‘งโˆˆฮ“(๐œ)||||๐‘“(๐‘ง)(1.34) for any ๐œโˆˆ๐•Š๐‘. [17, Theorem 1.3] will be very useful.

Theorem 1.12. Let ฮจ be a growth function. Then, for any ๐‘“โˆˆ๐ปฮจ, ๎€œ๐•Š๐‘ฮจ๎€ท||๐‘๐‘“||๎€ธ๐‘‘๐œŽโ‰ฒโ€–๐‘“โ€–ฮจ.(1.35) In particular, a holomorphic function ๐‘“ belongs to ๐ปฮจ if and only if ๐‘๐‘“ belongs to ๐ฟฮจ(๐•Š๐‘,๐œŽ).

From this theorem, we deduce the following one.

Theorem 1.13. Let ฮจ be a subharmonic-preserving growth function. Then for every ๐‘“โˆˆ๐ปฮจ, one has(1)lim๐‘Ÿโ†’1โˆซ๐•Š๐‘ฮจ(|๐‘“โˆ—โˆ’๐‘“๐‘Ÿ|)๐‘‘๐œŽ=0;(2)โ€–๐‘“โˆ—โ€–ฮจ=โ€–๐‘“โ€–ฮจ๐ปฮจ;(3)๐ปฮจ is separable. More precisely, the polynomials are dense in ๐ปฮจ.

Proof. Let ๐‘€rad๐‘“(๐œ)=sup0<๐‘Ÿ<1|๐‘“(๐‘Ÿ๐œ)| for ๐œโˆˆ๐•Š๐‘. Obviously, ๐‘€rad๐‘“โ‰ค๐‘๐‘“, hence โˆซ๐•Š๐‘ฮจ(๐‘€rad๐‘“)๐‘‘๐œŽโ‰ฒโ€–๐‘“โ€–ฮจ (Theorem 1.12). Since ฮจ is concave and vanishes at 0, we have ฮจ(|๐‘“โˆ—โˆ’๐‘“๐‘Ÿ|)โ‰คฮจ(|๐‘“โˆ—|)+ฮจ(|๐‘“๐‘Ÿ|)โ‰ค2ฮจ(๐‘€rad๐‘“). Now ฮจ(|๐‘“โˆ—โˆ’๐‘“๐‘Ÿ|(๐œ)) tends to 0 as ๐‘Ÿ goes to 1 for ๐œŽ-almost every ๐œโˆˆ๐•Š๐‘. By the dominated convergence theorem, (1) follows.
Then โ€–๐‘“โˆ—โ€–ฮจ=lim๐‘Ÿโ†’1โ€–๐‘“๐‘Ÿโ€–ฮจ and (2) comes from to the subharmonicity of ฮจ(|๐‘“|).
We proved in (1) that ๐‘“๐‘Ÿ tends to ๐‘“โˆ— in ๐ปฮจ for โ€–โ‹…โ€–ฮจ (hence for โ€–โ‹…โ€–ฮจ also). We approach every ๐‘“๐‘Ÿ uniformly on ๐”น๐‘ by its Taylor series to get the third assertion.

2. Carleson Embedding Theorems

2.1. Statements of the Results

For ๐œโˆˆ๐”น๐‘ and โ„Žโˆˆ]0,1], we define the nonisotropic โ€œballโ€™โ€™ of ๐”น๐‘ by๎€ฝ๐‘†(๐œ,โ„Ž)=๐‘งโˆˆ๐”น๐‘,๐‘‘(๐œ,๐‘ง)2๎€พ<โ„Ž(2.1) and its analogue in ๐”น๐‘ by๎‚†๐’ฎ(๐œ,โ„Ž)=๐‘งโˆˆ๐”น๐‘,๐‘‘(๐œ,๐‘ง)2๎‚‡<โ„Ž.(2.2) Let us also denote by๐‘„(๐œ,โ„Ž)=๐’ฎ(๐œ,โ„Ž)โˆฉ๐•Š๐‘,(2.3) the โ€œtrueโ€™โ€™ balls in ๐•Š๐‘. We have ๐œŽ(๐‘„(๐œ,โ„Ž))โ‰ˆโ„Ž๐‘ and ๐‘ฃ๐›ผ(๐‘†(๐œ,โ„Ž))โ‰ˆโ„Ž๐‘+๐›ผ+1 [22].

Let ๐œ‡ be a positive Borel measure on ๐”น๐‘ whose restriction to ๐•Š๐‘ is absolutely continuous with respect to ๐œŽ and let 0<๐‘โ‰ค๐‘ž<โˆž. By definition, ๐œ‡ is a (๐‘ž/๐‘)-Carleson measure if ๐œ‡(๐’ฎ(๐œ,โ„Ž))โ‰ค๐ถโ„Ž๐‘๐‘ž/๐‘, while it is a vanishing (๐‘ž/๐‘)-Carleson measure if ๐œ‡(๐’ฎ(๐œ,โ„Ž))=๐‘œ(โ„Ž๐‘๐‘ž/๐‘) when โ„Ž goes to 0. A variant of the well-known Carleson theorem for Hardy spaces [1, 3] ensures that the embedding ๐ป๐‘โ†ช๐ฟ๐‘ž(๐”น๐‘,๐œ‡) is bounded (resp., compact) if and only if ๐œ‡ is a (๐‘ž/๐‘)-Carleson measure (resp., a vanishing (๐‘ž/๐‘)-Carleson measure).

Similarly, we define the (๐›ผ,(๐‘ž/๐‘))-Bergman Carleson measures (resp., vanishing (๐›ผ,(๐‘ž/๐‘))-Bergman Carleson measures) for weighted Bergman spaces by ๐œ‡(๐‘†(๐œ,โ„Ž))โ‰ค๐ถโ„Ž(๐‘+๐›ผ+1)๐‘ž/๐‘ (resp., ๐œ‡(๐‘†(๐œ,โ„Ž))=๐‘œ(โ„Ž(๐‘+๐›ผ+1)๐‘ž/๐‘)). When ๐‘=๐‘ž, we just speak about ๐›ผ-Bergman Carleson measures (resp., vanishing ๐›ผ-Bergman Carleson measures). Ueki [7] showed that ๐ด๐‘๐›ผโ†ช๐ฟ๐‘ž(๐”น๐‘,๐œ‡) is bounded (resp., compact) if and only if ๐œ‡ is a (๐›ผ,(๐‘ž/๐‘))-Bergman Carleson measure (resp., a vanishing (๐›ผ,(๐‘ž/๐‘))-Carleson measure).

In the context of Hardy-Orlicz spaces (resp., weighted Bergman-Orlicz spaces) smaller than ๐ป1 (resp., ๐ด1๐›ผ) (i.e., when the defining function ฮจ 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 ๐ป1 (resp., ๐ด1๐›ผ), we state that the characterizations of the boundedness and compactness of ๐ปฮจ1โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) (resp., ๐ดฮจ1๐›ผโ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡)), where ฮจ1 and ฮจ2 are two growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing at least for large values of ๐‘ฅ (or equivalently ๐‘ฅโ†ฆฮจ2โˆ˜ฮจ1โˆ’1(๐‘ฅ)/๐‘ฅ nondecreasing, since ฮจ1 is increasing and does vanish except in 0), only depend on the growth of ฮจ2โˆ˜ฮจ1โˆ’1 at infinity.

Note that if ฮจ๐‘–(๐‘ฅ)=๐‘ฅ๐‘๐‘–, ฮจ๐‘–=ฮฆ๐‘๐‘– (with ฮฆ an Orlicz function), or ฮจ๐‘–(๐‘ฅ)=๐‘ฅ๐‘๐‘–log๐‘ž๐‘–(๐ถ+๐‘ฅ) with ๐‘2โ‰ฅ๐‘1, then ๐‘ฅโ†ฆฮจ2โˆ˜ฮจ1โˆ’1(๐‘ฅ)/๐‘ฅ is nondecreasing.

Theorem 2.1. Let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing. Let ๐œ‡ be a finite positive Borel measure on ๐”น๐‘ (whose restriction to ๐•Š๐‘ is absolutely continuous with respect to ๐œŽ). Then,(1)๐ปฮจ1 embeds into ๐ฟฮจ2(๐”น๐‘,๐œ‡) if and only if there exists some โ„Ž0โˆˆ(0,1) such that, for any โ„Žโˆˆ(0,โ„Ž0), 1๐œ‡(๐’ฎ(๐œ,โ„Ž))โ‰ฒฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๎€ธ(2.4) uniformly in ๐œโˆˆ๐•Š๐‘,(2)the embedding ๐ปฮจ1โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) is compact if and only if ๐œ‡(๐’ฎ(๐œ,โ„Ž))=๐‘œโ„Žโ†’0๎ƒฉ1ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๎€ธ๎ƒช(2.5) uniformly in ๐œโˆˆ๐•Š๐‘.A measure ๐œ‡ which satisfies (2.4) (resp., (2.5)) will be called a (ฮจ1,ฮจ2)-Carleson measure (resp., a vanishing (ฮจ1,ฮจ2)-Carleson measure).

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

Theorem 2.2. Let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing, and let ๐›ผ>โˆ’1. Let also ๐œ‡ be a finite positive Borel measure on ๐”น๐‘. Then,(1)๐ดฮจ1๐›ผ embeds into ๐ฟฮจ2(๐”น๐‘,๐œ‡) if and only if there exists โ„Ž0โˆˆ(0,1) such that, for any โ„Žโˆˆ(0,โ„Ž0), 1๐œ‡(๐‘†(๐œ,โ„Ž))โ‰ฒฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘+๐›ผ+1๎€ธ,(2.6) uniformly in ๐œโˆˆ๐•Š๐‘,(2)the embedding ๐ดฮจ1๐›ผโ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) is compact if and only if ๐œ‡(๐‘†(๐œ,โ„Ž))โ‰ค๐‘œโ„Žโ†’0๎ƒฉ1ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘+๐›ผ+1๎€ธ๎ƒช,(2.7) uniformly in ๐œโˆˆ๐•Š๐‘.A measure ๐œ‡ which satisfies (2.6) (resp., (2.7)) will be called a (๐›ผ,ฮจ1,ฮจ2)-Bergman-Carleson measure (resp., a vanishing (๐›ผ,ฮจ1,ฮจ2)-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 ๐›ผ>โˆ’1 and let 0<๐‘โ‰ค๐‘ž<1 and ฮจ1=ฮฆ๐‘, ฮจ2=ฮฆ๐‘ž as in (3) of Example 1.3. Let ๐œ‡ be a finite positive Borel measure on ๐”น๐‘ (whose restriction to ๐•Š๐‘ is absolutely continuous with respect to ๐œŽ) (resp., on ๐”น๐‘). Then,(1)๐ปฮฆ๐‘ (resp., ๐ดฮฆ๐‘๐›ผ) embeds into ๐ฟฮฆ๐‘ž(๐”น๐‘,๐œ‡) (resp., ๐ฟฮฆ๐‘ž(๐”น๐‘,๐œ‡)) if and only if ๐œ‡ is a (๐‘ž/๐‘)-Carleson measure (resp., a (๐›ผ,(๐‘ž/๐‘))-Bergman-Carleson measure),(2)the embedding ๐ปฮฆ๐‘โ†ช๐ฟฮฆ๐‘ž(๐”น๐‘,๐œ‡) (resp., ๐ดฮฆ๐‘๐›ผโ†ช๐ฟฮฆ๐‘ž(๐”น๐‘,๐œ‡)) is compact if and only if ๐œ‡ is a vanishing (๐‘ž/๐‘)-Carleson measure (resp., a vanishing (๐›ผ,(๐‘ž/๐‘)) Bergman-Carleson measure).

If ฮจ1=ฮจ2 (or equivalently if ฮจ1 and ฮจ2 are equivalent), we have the following.

Corollary 2.5. Let ๐›ผ>โˆ’1, and let ฮจ be subharmonic-preserving growth function. Let ๐œ‡ be a finite positive Borel measure on ๐”น๐‘ (whose restriction to ๐•Š๐‘ is absolutely continuous with respect to ๐œŽ) (resp., on ๐”น๐‘). Then,(1)๐ปฮจ (resp., ๐ดฮจ๐›ผ) embeds into ๐ฟฮจ(๐”น๐‘,๐œ‡) (resp., ๐ฟฮจ(๐”น๐‘,๐œ‡)) if and only if ๐œ‡ is a Carleson measure (resp., a ๐›ผ-Bergman-Carleson measure),(2)the embedding ๐ปฮจโ†ช๐ฟฮจ(๐”น๐‘,๐œ‡) (resp., ๐ดฮจ๐›ผโ†ช๐ฟฮจ(๐”น๐‘,๐œ‡)) is compact if and only if ๐œ‡ is a vanishing Carleson measure (resp., a vanishing ๐›ผ-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 ๐ปฮจ (resp., ๐ดฮจ๐›ผ), for ฮจ a subharmonic-preserving growth function, implies the convergence on every compact subset of ๐”น๐‘, but it stems from Proposition 1.10 (resp., Proposition 1.6).

Proposition 2.6. Let ๐›ผ>โˆ’1, let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions and let ๐œ‡ (resp., ๐œ‡) be a finite positive Borel measure on ๐”น๐‘ (resp., ๐”น๐‘) whose restriction to ๐•Š๐‘ is absolutely continuous with respect to ๐œŽ. One assume that ๐‘—๐œ‡โˆถ๐ปฮจ1โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) (resp., ๐‘—๐œ‡๐›ผโˆถ๐ดฮจ1๐›ผโ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡)) is well defined (hence bounded).(1)The two following assertions are equivalent:(a)the canonical embedding ๐‘—๐œ‡ (resp., ๐‘—๐œ‡๐›ผ) is compact;(b)every sequence in the unit ball of ๐ปฮจ1 (resp., ๐ดฮจ1๐›ผ), which is convergent to 0 uniformly on every compact subset of ๐”น๐‘, is convergent to 0 in ๐ฟฮจ2(๐œ‡) (resp., ๐ฟฮจ2(๐œ‡)).(2)If lim๐‘Ÿโ†’1โˆ’โ€–๐ผ๐‘Ÿโ€–=0 (resp., lim๐‘Ÿโ†’1โˆ’โ€–๐ผ๐‘Ÿ,๐›ผโ€–=0), where ๐ผ๐‘Ÿ(๐‘“)=๐‘“โ‹…๐œ’๐”น๐‘โงต๐‘Ÿ๐”น๐‘ (resp., ๐ผ๐‘Ÿ(๐‘“)=๐‘“โ‹…๐œ’๐”น๐‘โงต๐‘Ÿ๐”น๐‘), then the canonical embedding ๐‘—๐œ‡ (resp., ๐‘—๐œ‡๐›ผ) 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 ๐พ๐œ‡, associated to ๐œ‡ by๐พ๐œ‡(โ„Ž)=sup0<๐‘ก<โ„Žsup๐œโˆˆ๐•Š๐‘๐œ‡(๐’ฎ(๐œ,๐‘ก))๐‘ก๐‘,forโ„Žโˆˆ(0,1),(2.8) so that ๐œ‡ is a Carleson measure if and only if ๐พ๐œ‡(โ„Ž) is finite for some โ„Žโˆˆ(0,1).

Theorem 2.7 (see [12, Theorem 2.4]). There exist two constants ๎‚๐ถ>0 and ๐ถ>1 such that, for every ๐‘“ continuous on ๐”น๐‘ which admits boundary values almost everywhere on ๐•Š๐‘, and every finite positive Borel measure ๐œ‡ on ๐”น๐‘, one has ๐œ‡๎‚€๎‚†๐‘งโˆˆ๐”น๐‘||||โ‰ค๎‚,|๐‘ง|>1โˆ’โ„Ž,๐‘“(๐‘ง)>๐‘ก๎‚‡๎‚๐ถ๐พ๐œ‡๐‘(๐ถโ„Ž)๐œŽ๎€ท๎€ฝ๐‘“>๐‘ก๎€พ๎€ธ,(2.9) for every โ„Žโˆˆ(0,1/๐ถ) and every ๐‘ก>0.

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 ๐œ‡ be a finite positive Borel measure on ๐”น๐‘ and let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions. Let ๐ถโ‰ฅ1 be the constant appearing in Theorem 2.7. Assume that there exist ๐ด>0,๐œ‚>0 and โ„Ž๐ดโˆˆ(0,1/๐ถ) such that ๐พ๐œ‡(โ„Ž)โ‰ค๐œ‚1/โ„Ž๐‘ฮจ2๎€ท๐ดฮจ1โˆ’1๎€ท1/โ„Ž๐‘,๎€ธ๎€ธ(2.10) for every โ„Žโˆˆ(0,โ„Ž๐ด). Then, for every ๐‘“โˆˆ๐ปฮจ1(๐”น๐‘) such that โ€–๐‘“โ€–ฮจ1โ‰ค1 and every Borel subset ๐ธ of ๐”น๐‘, ๎€œ๐ธฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐œ‚๐•Š๐‘ฮจ1๎€ท๐‘๐‘“๎€ธ๐‘‘๐œŽ,(2.11) where โ‰ฒ involves a constant which is independent of ๐‘“, ๐œ‚, and ๐ธ.

Proof. Let ๐‘“โˆˆ๐ปฮจ1(๐”น๐‘) with โ€–๐‘“โ€–ฮจ1โ‰ค1. With the notations of the statement of the lemma and using Proposition 1.10, the proof of [12, Lemma 2.6] directly yields. ๎€œ๐ธฮจ2๎‚ต๐ด2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1||๐‘“||๎‚ถ๎€œ๐‘‘๐œ‡โ‰ค๐‘ฅ๐ด0ฮจโ€ฒ2+๐œ‚๎‚๐ถ(๐‘ )๐œ‡(๐ธ)๐‘‘๐‘ ๐ถ๐‘๎€œโˆž๐‘ฅ๐ดฮจ๎…ž2ฮจ(๐‘ )1๎€ท๎€ท๎€ท(๐ถ+1)๐ถ๐‘โˆ’1๎€ธ๎€ธ๐‘†๎€ธ/๐ดฮจ2๐‘(((๐ถ+1)/๐ถ)๐‘†)ร—๐œŽ๎‚ต๎‚ป๐‘“>2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1๐ด๐‘ ๎‚ผ๎‚ถ๐‘‘๐‘ ,(2.12) for some constant ๐‘ฅ๐ด which depends only on ๐ด. Now, since ฮจ1 and ฮจ2 are two concave growth functions, we have ฮจ๐‘–(๐‘ )โ‰ฒฮจ๐‘–(๐พ๐‘ )โ‰ฒฮจ๐‘–(๐‘ ),(2.13) where โ‰ฒ involve constants which only depend on ๐พ and ฮจ๐‘–. Therefore, (2.12) becomes ๎€œ๐ธฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐œ‚โˆž๐‘ฅ๐ดฮจ๎…ž2ฮจ(๐‘ )1(๐‘ )ฮจ2๐œŽ๐‘(๐‘ )๎‚ต๎‚ป๐‘“>2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1๐ด๐‘ ๎‚ผ๎‚ถ๐‘‘๐‘ .(2.14) Since ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing and since ฮจ1 and ฮจ2 and their derivatives do not vanish except in 0, we have, for any ๐‘ >0, ฮจ๎…ž1(๐‘ )ฮจ๎…ž2โ‰คฮจ(๐‘ )1(๐‘ )ฮจ2(๐‘ ).(2.15) It follows that ๎€œ๐ธฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐œ‚โˆž๐‘ฅ๐ดฮจ๎…ž1๐‘(๐‘ )๐œŽ๎‚ต๎‚ป๐‘“>2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1๐ด๐‘ ๎€œ๎‚ผ๎‚ถ๐‘‘๐‘ โ‰ฒ๐œ‡(๐ธ)+๐œ‚0+โˆžฮจ1๎‚ต๎‚ป๐ด๐‘๐‘“2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1๎€œ>๐‘ ๎‚ผ๎‚ถ๐‘‘๐‘ =๐œ‡(๐ธ)+๐œ‚๐•Š๐‘๐œ“1๎‚ต๐ด๐‘๐‘“2๐‘+1(๐ถ+1)๐ถ๐‘โˆ’1๎‚ถ๎€œ๐‘‘๐œŽโ‰ฒ๐œ‡(๐ธ)+๐œ‚๐•Š๐‘๐œ“1๎€ท๐‘๐‘“๎€ธ๐‘‘๐œŽ.(2.16)

We now can prove Theorem 2.1.

Proof of Theorem 2.1. (1) We assume that ๐œ‡ is a (ฮจ1,ฮจ2)-Carleson measure, and we intend to show that ๐‘—๐œ‡โˆถ๐ปฮจ1โ†ช๐ฟฮจ2(๐”น๐‘,๐œ‡) is well defined (hence bounded). We observe that (2.10) is satisfied for ๐ด=๐œ‚=1. Indeed we have, for any โ„Žโˆˆ(0,โ„Ž0), ๐œ‡(๐’ฎ(๐œ,โ„Ž))โ„Ž๐‘โ‰ฒ1/โ„Ž๐‘ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๎€ธ,(2.17) uniformly in ๐œโˆˆ๐•Š๐‘. Now, ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing (at least for large ๐‘ฅ) so that we can find โ„Ž1โˆˆ(0,min(1/๐ถ,โ„Ž0)) such that, for any โ„Žโˆˆ(0,โ„Ž1), ๐พ๐œ‡(โ„Ž)โ‰ฒ1/โ„Ž๐‘ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๎€ธ.(2.18) Therefore, we may and will apply Lemma 2.8 with ๐ธ=๐”น๐‘ to get ๎€œ๐”น๐‘ฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐•ŠNฮจ1๎€ท๐‘๐‘“๎€ธ๐‘‘๐œŽ.(2.19) We finish the proof of (1) using Theorem 1.12 and Inequality (1.7): ๎€œ๐”น๐‘ฮจ2๎€ท||๐‘“||๎€ธ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+โ€–๐‘“โ€–ฮจ1.(2.20)
For the converse, let ๐‘”๐‘Ž,๐‘=ฮฆโˆ’1((1/(1โˆ’|๐‘Ž|))๐‘/๐‘)๐‘“๐‘Ž,๐‘, where ๐‘“๐‘Ž,๐‘ is the test function introduced at the end of Section 1.2.2, with ๐‘ such that ฮจ1 is a growth function of order ๐‘. According to (1.31), ๐‘”๐‘Ž,๐‘ lies in the unit ball of ๐ปฮจ1. Let ๐ถ be the (finite) norm of ๐‘—๐œ‡. Using a classical computation which gives that |1โˆ’โŸจ๐‘ง,๐‘ŽโŸฉ|โ‰ค2(1โˆ’|๐‘Ž|) for any ๐‘งโˆˆ๐’ฎ(๐‘Ž/|๐‘Ž|,1โˆ’|๐‘Ž|), we get |๐‘“๐‘Ž,๐‘(๐‘ง)|โ‰ฅ1/4๐‘/๐‘ for such ๐‘ง, hence ๎€œ1โ‰ฅ๐”น๐‘ฮจ2๎ƒฉ||๐‘”๐‘Ž,๐‘||๐ถ๎ƒช๎‚ต๐’ฎ๎‚ต๐‘Ž๐‘‘๐œ‡โ‰ฅ๐œ‡ฮจ|๐‘Ž|,1โˆ’|๐‘Ž|๎‚ถ๎‚ถ2๎ƒฉ14๐‘/๐‘๐ถฮจ1โˆ’1๎ƒฉ๎‚ต1๎‚ถ1โˆ’|๐‘Ž|๐‘/๐‘๎ƒช๎ƒช(2.21) for any ๐‘Žโˆˆ๐”น๐‘. We may assume that ๐ถโ‰ฅ1 and since ฮจ2 is concave and vanishes at 0, it follows that ๐œ‡๎‚ต๐’ฎ๎‚ต๐‘Ž|๐‘Ž|,1โˆ’|๐‘Ž|๎‚ถ๎‚ถโ‰ค4๐‘/๐‘๐ถ1ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/(1โˆ’|๐‘Ž|)๐‘๎€ธ.(2.22)
(2) We turn to the compactness part. We first prove the sufficient part. Let us assume that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing and that ๐œ‡ is a vanishing (ฮจ1,ฮจ2)-Carleson measure. In particular, ๐œ‡ is a (ฮจ1,ฮจ2)-Carleson measure and we may apply Proposition 2.6. Then it is sufficient to prove that for every ๐œ€>0, there exists ๐‘Ÿ close enough to 1, such that โ€–๐ผ๐‘Ÿโ€–<๐œ€ (where ๐ผ๐‘Ÿโˆถ๐ปฮจ1โ†’๐ฟฮจ2(๐”น๐‘โงต๐‘Ÿ๐”น๐‘,๐œ‡)). We fix ๐œ€>0 and let ๐‘“ be in the unit ball of ๐ปฮจ2. As in the proof of the sufficient part of (1), since ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing, ๐œ‡ is a vanishing (ฮจ1,ฮจ2)-Carleson measure implying that ๐พ๐œ‡(โ„Ž)โ‰ฒ๐œ€1/โ„Ž๐‘ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๎€ธ,(2.23) for โ„Ž small enough. Then, we apply Lemma 2.8 with ๐œ‚=๐œ€ and ๐ธ=๐”น๐‘โงต๐‘Ÿ๐”น๐‘ to get the existence of ๐‘Ÿ close enough to 1 (independent of ๐‘“), such that โ€–โ€–๐ผ๐‘Ÿ(โ€–โ€–๐‘“)ฮจ2๎‚€โ‰ฒ๐œ‡๐”น๐‘โงต๐‘Ÿ๐”น๐‘๎‚๎€œ+๐œ€๐•Š๐‘ฮจ1๎€ท๐‘๐‘“๎€ธ๎‚€๐‘‘๐œŽโ‰ฒ๐œ‡๐”น๐‘โงต๐‘Ÿ๐”น๐‘๎‚+๐œ€โ€–๐‘“โ€–ฮจ1,(2.24) for any ๐œ€>0. Now, we may argue as in [12, Lemma 2.13] to prove that, under the assumption that ๐œ‡ is a (ฮจ1,ฮจ2)-Carleson measure, ๐œ‡(๐•Š๐‘)=0. Hence limsup๐‘Ÿโ†’1โ€–โ€–๐ผ๐‘Ÿโ€–โ€–(๐‘“)ฮจ2โ‰ฒ๐œ€โ€–๐‘“โ€–ฮจ1,(2.25) for any ๐œ€>0, which gives (2).
For the converse, we assume that ๐‘—๐œ‡ is compact but that ๐œ‡ is not a vanishing (ฮจ1,ฮจ2)-Carleson measure, that is there exist ๐œ€0>0, a sequence (โ„Ž๐‘›)๐‘› decreasing to 0 and a sequence (๐œ๐‘›)๐‘›โŠ‚๐•Š๐‘ such that ๐œ‡๎€ท๐’ฎ๎€ท๐œ๐‘›,โ„Ž๐‘›๎€ธ๎€ธโ‰ฅ๐œ€01ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๐‘›๎€ธ.(2.26) Now, let ๐‘”๐‘Ž๐‘›,๐‘=ฮฆโˆ’1((1/(1โˆ’|๐‘Ž๐‘›|))๐‘/๐‘)๐‘“๐‘Ž๐‘›,๐‘ be as in the proof of the necessity in the boundedness part with ๐‘Ž๐‘›=(1โˆ’โ„Ž๐‘›)๐œ๐‘› and ๐‘ such that ฮจ1 is growth function of order ๐‘. Since (๐‘”๐‘Ž๐‘›,๐‘)๐‘› is bounded in ๐ปฮจ1 and converges uniformly on every compact subset of ๐”น๐‘, then we must have โ€–๐‘—๐œ‡(๐‘”๐‘Ž๐‘›,๐‘)โ€–ฮจ2โ†’๐‘›โ†’โˆž0, because of Proposition 2.6. But, for ๐‘งโˆˆ๐’ฎ(๐œ๐‘›,โ„Ž๐‘›), we saw that ||๐‘”๐‘Ž๐‘›,๐‘||โ‰ฅ1(๐‘ง)4๐‘/๐‘ฮจ1โˆ’1๎‚ต1โ„Ž๐‘๐‘›๎‚ถ.(2.27) Since ฮจ2 is a growth function of order, let say ๐‘ž, we have, for any ๐‘กโˆˆ(0,1), ฮจ2(๐‘ฆ/๐‘ก)โ‰ฅฮจ2(๐‘ฆ)/๐‘ก๐‘ž. Hence, we get ๎€œ๐”น๐‘ฮจ2๎ƒฉ4๐‘๐‘ž/๐‘||๐‘”๐‘Ž๐‘›,๐‘||๐œ€01/๐‘ž๎ƒช๐‘‘๐œ‡โ‰ฅฮจ2๎ƒฉ1๐œ€01/๐‘žฮจ1โˆ’1๎€ท1/โ„Ž๐‘๐‘›๎€ธ๎ƒช๐œ‡๎€ท๐’ฎ๎€ท๐œ๐‘›,โ„Ž๐‘›โ‰ฅฮจ๎€ธ๎€ธ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๐‘›๎€ธ๐œ€0๐œ€01ฮจ2โˆ˜ฮจ1โˆ’1๎€ท1/โ„Ž๐‘๐‘›๎€ธ=1.(2.28) Therefore, โ€–๐‘—๐œ‡(๐‘”๐‘Ž๐‘›,๐‘)โ€–ฮจ2โ‰ฅ๐œ€01/๐‘ž/4๐‘๐‘ž/๐‘, which is a contradiction.

2.2.2. Proof of Theorem 2.2

We introduce the following function:๐พ๐œ‡,๐›ผ(โ„Ž)=sup0<๐‘ก<โ„Žsup๐œโˆˆ๐•Š๐‘๐œ‡(๐‘†(๐œ,๐‘ก))๐‘ก๐‘+๐›ผ+1.(2.29) 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 ๎‚๐ถ>0 such that, for every ๐‘“ continuous on ๐”น๐‘ and every positive finite Borel measure ๐œ‡ on ๐”น๐‘, one has ๐œ‡๎€ท๎€ฝ๐‘งโˆˆ๐”น๐‘||||โ‰ค๎‚,|๐‘ง|>1โˆ’โ„Ž,๐‘“(๐‘ง)>๐‘ก๎€พ๎€ธ๐ถ๐พ๐œ‡,๐›ผ(2โ„Ž)๐‘ฃ๐›ผฮ›๎€ท๎€ฝ๐‘“,>๐‘ก๎€พ๎€ธ(2.30) for every โ„Žโˆˆ(0,1/2) and every ๐‘ก>0.

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 ๐œ‡ be a finite positive Borel measure on ๐”น๐‘ and let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions. Assume that there exist ๐ด>0, ๐œ‚>0, and โ„Ž๐ดโˆˆ(0,1/2) such that ๐พ๐œ‡,๐›ผ(โ„Ž)โ‰ค๐œ‚1/โ„Ž๐‘+1+๐›ผ๐œ“2๎€ท๐ดฮจ1โˆ’1๎€ท1/โ„Ž๐‘+1+๐›ผ,๎€ธ๎€ธ(2.31) for every โ„Žโˆˆ(0,โ„Ž๐ด). Then, for every ๐‘“โˆˆ๐ดฮจ1๐›ผ(๐”น๐‘) with โ€–๐‘“โ€–๐ดฮจ1๐›ผโ‰ค1 and every Borel subset ๐ธ of ๐”น๐‘, one has ๎€œ๐ธฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐œ‚๐”น๐‘ฮจ1๎€ทฮ›๐‘“๎€ธ๐‘‘๐‘ฃ๐›ผ,(2.32) where โ‰ฒ involves constants which are independent of ๐‘“, ๐œ‚, and ๐ธ.

We now prove Theorem 2.2.

Proof of Theorem 2.2. Let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing.
(1) We first assume that ๐œ‡ is a (๐›ผ,ฮจ1,ฮจ2)-Bergman Carleson measure, which implies, as in the beginning of the proof of (1) of Theorem 2.1, that condition (2.31) with ๐ด=๐œ‚=1 is fulfilled. Applying (2.31) to ๐‘“, with ๐ธ=๐”น๐‘, gives ๎€œ๐”น๐‘ฮจ2๎€ท||๐‘“||๎€ธ๎€œ๐‘‘๐œ‡โ‰ฒ๐œ‡(๐ธ)+๐”น๐‘ฮจ1๎€ทฮ›๐‘“๎€ธ๐‘‘๐‘ฃ๐›ผ.(2.33) 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 ๐‘”๐‘Ž,๐›ผ,๐‘=ฮฆโˆ’1((1/(1โˆ’|๐‘Ž|))(๐‘+๐›ผ+1)/๐‘)๐‘“๐‘Ž,๐›ผ,๐‘, where ๐‘“๐‘Ž,๐›ผ,๐‘ has been introduced in Section 1.2.1, with ๐‘ such that ฮจ1 is a growth function of order ๐‘.
(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 ๐‘“โˆˆ๐ดฮจ1๐›ผ, with ๐œ‚=๐œ€ and ๐ธ=๐”น๐‘โงต๐‘Ÿ๐”น๐‘ to show that โ€–๐‘“.๐œ’๐”น๐‘โงต๐‘Ÿ๐”น๐‘โ€–ฮจ2 tends to 0, as ๐‘Ÿ tends to 1. Proposition 2.6 then shows that ๐ดฮจ1๐›ผโ†ช๐ฟฮจ2(๐œ‡) 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 ๐ปฮจ and ๐ดฮจ๐›ผ. As usual, for ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ holomorphic, ๐ถ๐œ™ will be seen as an embedding operator. To do so, we define the pullback measures ๐œ‡๐œ™ and ๐œ‡๐œ™,๐›ผ of, respectively, ๐œŽ and ๐‘ฃ๐›ผ under ๐œ™:๐œ‡๐œ™๎€ท๐œ™(๐ธ)=๐œŽโˆ—โˆ’1(๐ธ)โˆฉ๐•Š๐‘๎€ธ,(3.1) where ๐œ™โˆ— is the radial limit almost everywhere of ๐œ™ and ๐ธ is any Borel subset of ๐”น๐‘. And๐œ‡๐œ™,๐›ผ(๐ธ)=๐‘ฃ๐›ผ๎€ท๐œ™โˆ’1๎€ธ(๐ธ)(3.2) for any Borel subset ๐ธ of ๐”น๐‘.

By a classical formula for pull-back measures,โ€–โ€–๐‘—๐œ‡๐œ™โ€–โ€–(๐‘“)ฮจ=โ€–โ€–๐ถ๐œ™โ€–โ€–(๐‘“)ฮจ๎‚€โ€–โ€–๐‘—resp.,๐œ‡๐œ™,๐›ผโ€–โ€–(๐‘“)๐›ผ,ฮจ=โ€–โ€–๐ถ๐œ™โ€–โ€–(๐‘“)๐›ผ,ฮจ๎‚(3.3) for any growth function ฮจ and every polynomial ๐‘“ (resp., for any ๐‘“โˆˆ๐ดฮจ๐›ผ). Then we extend this equality to the whole space ๐ปฮจ by density of polynomials in ๐ปฮจ (Theorem 1.13) and using Cauchyโ€™s formula.

Remark 3.1. Observe that it is not true that ๐‘—๐œ‡๐œ™(๐‘“)=๐ถ๐œ™(๐‘“) for any ๐‘“โˆˆ๐ปฮจ but it is for any polynomial ๐‘“. Such an equality holds for every function in ๐ดฮจ๐›ผ.

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

Proposition 3.2. Let ๐›ผ>โˆ’1 and let ฮจ1 and ฮจ2 be as in Proposition 2.6. Let also ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ be holomorphic. ๐ถ๐œ™ is compact from ๐ปฮจ1 into ๐ปฮจ2 (resp., ๐ดฮจ1๐›ผ into ๐ดฮจ2๐›ผ) if and only if, for every bounded sequence (๐‘“๐‘›)๐‘›โŠ‚๐ปฮจ1 (resp., (๐‘“๐‘›)๐‘›โŠ‚๐ดฮจ1๐›ผ) which converges to 0 on every compact of ๐”น๐‘,โ€–๐ถ๐œ™(๐‘“)โ€–ฮจ2 (resp., โ€–๐ถ๐œ™(๐‘“)โ€–ฮจ2,๐›ผ) 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 ๐ถ๐œ™ on ๐ปฮจ and ๐ดฮจ๐›ผ.

Theorem 3.3. Let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing. Let also ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ be holomorphic. Then,(1)๐ถ๐œ™ is bounded from ๐ปฮจ1 into ๐ปฮจ2 if and only if ๐œ‡๐œ™ is a (ฮจ1,ฮจ2)-Carleson measure,(2)๐ถ๐œ™ is compact from ๐ปฮจ1 into ๐ปฮจ2 if and only if ๐œ‡๐œ™ is a vanishing (ฮจ1,ฮจ2)-Carleson measure.

For Bergman-Orlicz spaces, we have the following.

Theorem 3.4. Let ๐›ผ>โˆ’1 and let ฮจ1 and ฮจ2 be two subharmonic-preserving growth functions such that ๐‘ฅโ†ฆฮจ2(๐‘ฅ)/ฮจ1(๐‘ฅ) is nondecreasing. Let also ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ be holomorphic. Then,(1)๐ถ๐œ™ is bounded from ๐ดฮจ1๐›ผ into ๐ดฮจ2๐›ผ if and only if ๐œ‡๐œ™,๐›ผ is a (๐›ผ,ฮจ1,ฮจ2)-Bergman Carleson measure,(2)๐ถ๐œ™ is compact from ๐ดฮจ1๐›ผ into ๐ดฮจ2๐›ผ if and only if ๐œ‡๐œ™,๐›ผ is a vanishing (๐›ผ,ฮจ1,ฮจ2)-Bergman Carleson measure.

We immediately deduce the following corollary, which reminds a similar result when the function ฮจ is an Orlicz function satisfying the ฮ”2-Condition (see [11, 12]).

Corollary 3.5. Let ๐›ผ>โˆ’1 and let ฮจ be a subharmonic preserving growth function. Let also ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ be holomorphic. Then,(1)C๐œ™ is bounded (resp., compact) on ๐ปฮจ if and only if it is bounded (resp., compact) on one (or equivalently every) ๐ป๐‘, if and only if ๐œ‡๐œ™ is a Carleson measure (resp., a vanishing Carleson measure),(2)๐ถ๐œ™ is bounded (resp., compact) on ๐ดฮจ๐›ผ if and only if it is bounded (resp., compact) on one (or equivalently every) ๐ด๐‘๐›ผ, if and only if ๐œ‡๐œ™ 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 ๐›ผ>โˆ’1, let 0<๐‘โ‰ค๐‘ž<โˆž and ฮจ1=ฮฆ๐‘, and ฮจ2=ฮฆ๐‘ž as in (3) of Example 1.3. Let also ๐œ™โˆถ๐”น๐‘โ†’๐”น๐‘ be holomorphic. Then,(1)๐ถ๐œ™ is bounded (resp., compact) from ๐ปฮจ1 to ๐ปฮจ2 if and only if it is bounded (resp., compact) from ๐ป๐‘ to ๐ป๐‘ž, if and only if ๐œ‡๐œ™ is a ๐‘ž/๐‘-Carleson measure (resp., a vanishing ๐‘ž/๐‘-Carleson measure),(2)๐ถ๐œ™ is bounded (resp., compact) from ๐ดฮจ1๐›ผ to ๐ดฮจ2๐›ผ if and only if it is bounded (resp., compact) from ๐ด๐‘๐›ผ to ๐ด๐‘ž๐›ผ, if and only if ๐œ‡๐œ™ is a ๐‘ž/๐‘-Bergman-Carleson measure (resp., a vanishing ๐‘ž/๐‘-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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