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

Abstract

We characterize those measures 𝜇 for which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) space 𝐻^Ψ1 (resp., 𝐴^Ψ1_𝛼) of the unit ball of ℂ^𝑁 embeds boundedly or compactly into the Orlicz space 𝐿^Ψ2(𝔹_𝑁,𝜇) (resp., 𝐿^Ψ2(𝔹_𝑁,𝜇)), when the defining functions Ψ₁ and Ψ₂ are growth functions such that 𝐿¹⊂𝐿^Ψ𝑗 for 𝑗∈{1,2}, and such that Ψ₂/Ψ₁ is nondecreasing. We apply our result to the characterization of the boundedness and compactness of composition operators from 𝐻^Ψ1 (resp., 𝐴^Ψ1_𝛼) into 𝐻^Ψ2 (resp., 𝐴^Ψ2_𝛼).

1. Introduction and Preliminaries

1.1. Introduction

Let 𝔹_𝑁={𝑧=(𝑧₁,…,𝑧_𝑁)∈ℂ^𝑁∶|𝑧|²=∑^𝑁𝑖=1|𝑧_𝑖|²<1} and 𝕊_𝑁=𝜕𝔹_𝑁, 𝑁≥1, denoting, respectively, the unit ball and the unit sphere of ℂ^𝑁. For 𝑁=1, we denote by 𝔻 the unit disc of the complex plane.

For a large class of spaces 𝑋 of holomorphic functions in the unit disc or the unit ball, characterizations of the boundedness and compactness of the canonical embedding 𝑋↪𝐿^𝑝(𝜇) 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 𝑋=𝐻^𝑝(𝔻), Carleson [1] proved that 𝐻^𝑝(𝔻)↪𝐿^𝑝(𝜇) if and only if the finite positive Borel measure 𝜇 on 𝔻 (or 𝔻) 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 𝜇 such that 𝐻^𝑝(𝔻)↪𝐿^𝑞(𝜇), with 0<𝑝≤𝑞<∞, in terms of (𝑞/𝑝)-Carleson measures. For the unweighted and weighted Bergman spaces 𝐴^𝑝𝛼(𝔹_𝑁), 𝑁≥1, similar results were obtained by Cima and Wogen [5], Luecking [6], Ueki [7]. We recall also that the compactness of 𝑋↪𝐿^𝑝(𝜇) was also characterized in the previous cases, in terms of vanishing Carleson’s type measures. It is usual to assume that when the measure 𝜇 is defined on 𝔹_𝑁, then 𝜇_∣𝕊𝑁 is absolutely continuous with respect to 𝕊_𝑁. 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 𝐻^𝑝(𝔹_𝑁)↪𝐿^𝑞(𝜇) are not always satisfied and just depend on the ratio 𝑞/𝑝 for 0<𝑝≤𝑞<∞; in particular, when 𝑝=𝑞, they are independent of 𝑝. Now, since the restriction to 𝕊_𝑁 of the finite positive Borel 𝜇 is assumed to be absolutely continuous with respect to the Lebesgue measure, then it is trivial that 𝐻^∞↪𝐿^∞(𝜇) always holds. On the contrary, the compactness of this inclusion implies strong condition on 𝜇. This suggests to think about this question when the space 𝑋 is intercalated between every 𝐻^𝑝 and 𝐻^∞.

This motivation is reinforced by a second observation: if the measure 𝜇_𝜙 is the pull-back measure of the invariant-rotation measure on 𝕊_𝑁 under a holomorphic map 𝜙∶𝔹_N→𝔹_𝑁, 𝑁≥1, then 𝜇_𝜙 is always a Carleson measure when 𝑁=1 (this is the Littlewood Subordination Principle, see [8]), but it is not systematic for 𝑁>1. For the compactness, there is still a big gap between 𝐻^𝑝 and 𝐻^∞ for any 𝑁≥1. This observation is directly connected to the study of composition operators 𝐶_𝜙 on 𝐻^𝑝, which are defined by 𝐶_𝜙(𝑓)=𝑓∘𝜙 for 𝑓∈𝐻(𝔹_𝑁), and which may be seen as the embedding operators 𝐻^𝑝↪𝐿^𝑝(𝜇_𝜙). By the way, this leads the authors of [9, 10] to state Carleson theorems for Hardy-Orlicz and Bergman-Orlicz spaces (resp., denoted by 𝐻^Ψ(𝔻) and 𝐴^Ψ(𝔻)) in the unit disc, when the defining function Ψ is an Orlicz function. These spaces appear as good candidates for generalizing 𝐻^𝑝 and 𝐴^𝑝 spaces, 1≤𝑝<∞, and for covering the gap with 𝐻^∞. 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 Ψ. Indeed, he showed that, when 𝜇=𝜇_𝜙 is the pull-back measure under a holomorphic self-map 𝜙 of 𝔹_𝑁, then 𝐻^Ψ always embeds into 𝐿^Ψ(𝜇) whenever Ψ satisfied a fast growth condition (namely, the Δ²-Condition, that implies Ψ(𝑥)≥𝑒^𝑥 for large value of 𝑥 and means that we are close to 𝐻^∞). Similarly, it was shown in [13] that if 𝐻^Ψ compactly embeds into 𝐿^Ψ(𝜇) for every Ψ, then 𝐻^∞↪𝐿^∞(𝜇) compactly (the converse easily holds). We mention that there is no Ψ such that 𝐻^Ψ↪𝐿^Ψ(𝜇) if and only if 𝐻^∞↪𝐿^∞(𝜇) [14], and that such previous results are not true for arbitrary measure 𝜇. Yet, a link has been made between the involved Carleson conditions and the type of growth of the Orlicz function Ψ. This is also strengthened by the fact that if the Orlicz function Ψ is dominated by a power function (exactly Ψ satisfies the Δ₂-Condition), then 𝐻^Ψ↪𝐿^Ψ(𝜇) boundedly (resp., compactly) if and only if 𝐻^𝑝↪𝐿^𝑝(𝜇), that is, if and only if 𝜇 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 𝐻¹(𝔹_𝑁). It seems that nothing has been done in this direction, except for explicit functions Ψ. In particular, the second author gave a necessary and sufficient condition for the inclusion 𝐻^Ψ(𝔹_𝑁)↪𝐿^𝑞(𝜇) to be bounded, when Ψ(𝑡)=(𝑡/log(𝑒+𝑡))^𝑠, 0<𝑠≤1, and 1≤𝑞<∞ [15]. Moreover, [16] characterized Carleson measures for 𝐴^Ψ𝛼(𝔻), where Ψ=log^𝑝+, 1≤𝑝<∞, that is, when 𝐴^Ψ𝛼(𝔻) is the area Nevanlinna space. These measures reveal to be those 𝜇 for which 𝐴^𝑝𝛼↪𝐿^𝑝(𝜇) 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 Ψ, 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 𝜇 such that 𝐻^Ψ1(𝔹_𝑁)↪𝐿^Ψ2(𝔹_𝑁,𝜇) (resp., 𝐴^Ψ1_𝛼(𝔹_𝑁)↪𝐿^Ψ2(𝔹_𝑁,𝜇)) is bounded or compact, when Ψ₁ and Ψ₂ are two growth functions (i.e., for which 𝐻¹⊂𝐻^Ψ𝑖⊂𝐻^𝑝 (resp., 𝐴^1𝛼⊂𝐴^Ψ𝑖_𝛼⊂𝐴^𝑝𝛼) for some 𝑝, 𝑖,𝑗∈{1,2}), and such that Ψ₂  grows faster than Ψ₁. It appears that if Ψ₁=Ψ₂, then such measures are exactly those which are Carleson (resp., Bergman-Carleson) measures for some 0<𝑝<1. For the Bergman-Orlicz case, these results let one think that, between the area Nevanlinna class [16] and 𝐴^1𝛼, there is no difference regarding to Carleson theorems, whenever the defining functions Ψ share some natural properties. This is in contrast with what happens between 𝐴^1𝛼 and 𝐻^∞.

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 𝑧,𝑤∈ℂ^𝑁, the euclidean inner product of 𝑧 and 𝑤 will be denoted by ⟨𝑧,𝑤⟩, that is, ⟨𝑧,𝑤⟩=∑^𝑁𝑖=1𝑧_𝑖𝑤_𝑖; the notation |⋅| will stand for the associated norm, as well as for the modulus of a complex number.

𝜎 will stand for the invariant-rotation measure on the unit sphere. For 𝛼>−1,𝑣_𝛼 will be the measure on 𝔹_𝑁 defined by 𝑑𝑣_𝛼=𝑐_𝛼(1−|𝑧|)^𝛼𝑑𝑣, where 𝑣 is the Lebesgue measure on 𝔹_𝑁 and 𝑐_𝛼 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 𝐻^𝜓 (resp., 𝐴^𝜓𝛼) instead of 𝐻^𝜓(𝔹_𝑁) (resp., 𝐴^𝜓𝛼(𝔹_𝑁)).

1.2. Preliminaries: Menagerie of Spaces

Let Ψ∶[0,∞)→[0,∞) be a continuous nondecreasing function which vanishes and is continuous at 0. Given a probabilistic space (Ω,ℙ), we define the Orlicz class 𝐿^Ψ(Ω,ℙ) as the set of all (equivalence classes of) measurable functions 𝑓 on Ω such that ∫_ΩΨ(|𝑓|/𝐶)𝑑ℙ<∞ for some 0<𝐶<∞. We use to define the Morse-Transue space 𝑀^Ψ(Ω,ℙ) by𝑀^Ψ(Ω,ℙ)=𝑓∶Ω⟶ℂmeasurable;_ΩΨ||𝑓||𝐶<∞forany𝐶>0,(1.1) and we also introduce the following set:ℒ^Ψ(Ω,ℙ)=𝑓∶Ω⟶ℂmeasurable;_ΩΨ||𝑓||<∞.(1.2) In general, these three sets are not vector spaces and do not coincide, but we trivially have𝑀^Ψ(Ω,ℙ)⊂ℒ^Ψ(Ω,ℙ)⊂𝐿^Ψ(Ω,ℙ).(1.3) We also define the Luxembourg gauge on 𝐿^Ψ(Ω,ℙ) by‖𝑓‖_Ψ=inf𝜆>0,_ΩΨ||𝑓||𝜆≤1.(1.4) This functional is homogeneous and is 0 if and only if 𝑓=0ℙ−a.e., but it is not subadditive a priori.

We say that two functions Ψ₁ and Ψ₂ as above are equivalent if there exists some constant 𝑐 such that𝑐Ψ₁(𝑐𝑥)≤Ψ₂(𝑥)≤𝑐⁻¹Ψ₁𝑐⁻¹𝑥,(1.5) for any 𝑥 large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.

In order to define a good topology on 𝐿^Ψ(Ω,ℙ) and to get properties convenient for our purpose, we will assume that Ψ satisfies the following definition.

Definition 1.1. Let 0<𝑝≤1. We say that Ψ∶[0,∞)→[0,∞) is a growth function of order 𝑝 if it satisfies the two following conditions:(1)Ψ is of lower type 𝑝, that is, Ψ(𝑦𝑥)≤𝑦^𝑝Ψ(𝑥) for any 0<𝑦≤1 and at least for 𝑥 large enough;(2)𝑥↦Ψ(𝑥)/𝑥 is nonincreasing, at least for every 𝑥 large enough.

We shall say that 𝜓 is a growth function if it is a growth function of order 𝑝 for some 0<𝑝≤1.

In particular, a growth function Ψ is equivalent to the function 𝑥↦∫^𝑥0(Ψ(𝑠)/𝑠)𝑑𝑠 which is concave (see [17]). Now, for such a concave growth function of order 𝑝,𝐿¹(Ω‖,𝑓ℙ‖)_Ψ⊂𝑀^Ψ(Ω,ℙ)=ℒ^Ψ(Ω,ℙ)=𝐿^Ψ(Ω,ℙ)⊂𝐿^𝑝(Ω,ℙ),(1.6)≲min_ΩΨ||𝑓||𝑑ℙ,_ΩΨ||𝑓||𝑑ℙ^𝑝,(1.7) while‖𝑓‖^Ψ∶=_ΩΨ||𝑓||𝑑ℙ≲max‖𝑓‖_Ψ,‖𝑓‖^𝑝Ψ.(1.8) Moreover, if we define 𝑑_Ψ(𝑓,𝑔)=‖𝑓−𝑔‖_Ψ (resp., 𝑑^Ψ(𝑓,𝑔)=‖𝑓−𝑔‖^Ψ), then 𝑑_Ψ and 𝑑^Ψ (that we will simply denote by ‖⋅‖_Ψ and ‖⋅‖^Ψ) are two equivalent metrics on 𝐿^Ψ(Ω,ℙ) 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 Δ₂-condition, that is, Ψ(2𝑥)≤𝐾Ψ(𝑥) for some 𝐾>1 and 𝑥 large enough. This condition is very classical when the function Ψ 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 Ψ is such that Ψ(|𝑓|) is subharmonic when 𝑓 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)  Ψ₁(𝑥)=𝑥^𝑝 for 0<𝑝≤1 and any 𝑥≥𝑥₀.
(2)  Ψ₂(𝑥)=𝑥^𝑝log^𝑞(𝐶+𝑥) (for 𝑥≥𝑥₀) with 𝐶>0 large enough, 0<𝑝≤1 and 𝑞≥1.
(3)  Ψ₃(𝑥)=Φ^𝑝(𝑥) at least for any 𝑥 large, where Φ is an Orlicz function (see Remark 1.2) and 𝑝>0 is such that Φ^𝑝(𝑥)/𝑥 is nonincreasing. Note that in this case, Ψ(|𝑓|) is subharmonic whenever 𝑓 is holomorphic, because we have the following.

Proposition 1.4. Let Ψ be as in (3) above. There exists a convex function 𝜓 such that Ψ(𝑥)=𝜓(𝑥^𝑝) for any 𝑥>0 large enough.

Proof. Let 𝑝>0 and Φ be an Orlicz function such that Ψ(𝑥)=Φ^𝑝(𝑥) for any 𝑥≥0. Since Ψ is (assumed to be) bijective, we can define the function 𝜓 by 𝜓(𝑥^𝑝)=Φ^𝑝(𝑥) for every 𝑥. Now, using that Φ is convex, 𝜓 is convex since 𝜓(𝑥)𝑥=Φ𝑥^1/𝑝𝑥^1/𝑝^𝑝≤Φ𝑦^1/𝑝𝑦^1/𝑝^𝑝=𝜓(𝑦)𝑦,(1.9) for any 0≤𝑥≤𝑦.

It follows that Ψ(|𝑓|)=𝜓(|𝑓|^𝑝) is subharmonic (for 𝑓 holomorphic).

The following lemma gives an upper estimate of the Luxembourg norm of a function in 𝐿^∞(Ω,ℙ).

Lemma 1.5. Let (Ω,ℙ) be a probabilistic space and let Ψ be a growth function of order 𝑝. For any 𝑓∈𝐿^∞(Ω,ℙ), one has ‖𝑓‖_Ψ≤‖𝑓‖^∞Ψ⁻¹‖𝑓‖_∞/‖𝑓‖^𝑝𝑝.(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 ‖𝑓‖_∞=1. For every 𝐶>0, one has the following, using that Ψ is of lower type 𝑝: _ΩΨ||𝑓||𝐶𝑑ℙ≤_Ω||𝑓||^𝑝Ψ1𝐶𝑑ℙ≤‖𝑓‖^𝑝𝑝Ψ1𝐶.(1.11) Then the last expression is less than or equal to 1 if and only if 𝐶≥1/Ψ⁻¹(1/‖𝑓‖^𝑝𝑝).

1.2.1. Large Bergman-Orlicz Spaces

For Ψ a subharmonic-preserving growth function and 𝛼>−1, the weighted Bergman-Orlicz space 𝐴^Ψ𝛼 of the ball consists of those holomorphic functions on 𝔹_𝑁 which belongs to the Orlicz space 𝐿^Ψ(𝔹_𝑁,𝑣_𝛼). To avoid further confusion, we will denote by ‖⋅‖_𝛼,Ψ the corresponding Luxembourg (quasi)norm and by ‖⋅‖^Ψ𝛼 the quantity ∫_𝔹𝑁Ψ(|𝑓|)𝑑𝑣_𝛼. 𝐴^Ψ𝛼 is metric space for the distance 𝑑_𝛼,Ψ or 𝑑^Ψ𝛼 defined by, respectively, 𝑑_𝛼,Ψ(𝑓,𝑔)=‖𝑓−𝑔‖_𝛼,Ψ and 𝑑^Ψ𝛼(𝑓,𝑔)=‖𝑓−𝑔‖^Ψ𝛼. If Ψ(𝑡)=𝑡^𝑝, then we recover the usual weighted Bergman space 𝐴^𝑝𝛼. One checks that we have the followings inclusions:𝐴^1𝛼⊂𝐴^Ψ𝛼⊂𝐴^𝑝𝛼,(1.12) whenever Ψ is a growth function of order 𝑝.

It seems important to us to mention that a linear operator 𝑇 from 𝐴^Ψ1_𝛼 to 𝑋 with 𝑋=𝐿^Ψ2(𝔹_𝑁,𝑣_𝛼) or 𝑋=𝐴^Ψ2_𝛼, where Ψ₁ and Ψ₂ 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 𝐶>0 such thatmax‖𝑇(𝑓)‖_𝛼,Ψ2,‖𝑇(𝑓)‖^Ψ2_𝛼≤𝐶,(1.13) for any 𝑓∈𝐴^Ψ1_𝛼 such that min(‖𝑓‖_𝛼,Ψ1,‖𝑓‖^Ψ1_𝛼)≤1. If 𝑋=ℂ, then 𝑇 is bounded if and only if |𝑇(𝑓)|≤𝐶 for 𝑓 as previously.

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

Proposition 1.6. Let 𝛼>−1 and let Ψ be a subharmonic-preserving growth function. For any 𝑎∈𝔹_𝑁 and any 𝑓∈𝐴^Ψ𝛼, one has ||𝑓(𝑎)||≤Ψ⁻¹1−2|𝑎|^{(𝑁+𝛼+1)}‖𝑓‖^𝛼,Ψ.(1.14)

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

Corollary 1.7. 𝐴^Ψ𝛼, endowed with 𝑑_𝛼,Ψ or 𝑑^Ψ𝛼, is a complete metric space.

Let 𝑝>0 and 𝛼>−1. For 𝑎∈𝔹_𝑁, we introduce the following “test’’ function 𝑓_{𝑎,𝛼,𝑝} defined by𝑓_{𝑎,𝛼,𝑝}=1−|𝑎|²(1−⟨𝑧,𝑎⟩)²^{(𝑁+𝛼+1)/𝑝}.(1.15)|𝑓_{𝑎,𝛼,𝑝}|^𝑝 is nothing but the Berezin kernel, hence ‖𝑓_{𝑎,𝛼,𝑝}‖_∞=((1+|𝑎|)/(1−|𝑎|))^{(𝑁+𝛼+1)/𝑝} while ‖𝑓_{𝑎,𝛼,𝑝}‖_𝑝=1. Then, as a consequence of Lemma 1.5, we have‖‖𝑓_{𝑎,𝛼,𝑝}‖‖_𝛼,Ψ≤21−|𝑎|^{(𝑁+𝛼+1)/𝑝}1Ψ⁻¹(2/1−|𝑎|)^{(𝑁+𝛼+1)},(1.16) whenever Ψ is a growth function of order 𝑝. 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 𝐴^Ψ𝛼. The definition needs to introduce the sets 𝑄(𝜁,ℎ), defined by𝑄(𝜁,ℎ)=𝑧∈𝕊_𝑁,||1−⟨𝑧,𝜁⟩||<ℎ,(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 𝑀>0 such that for any 0<𝑟<1, one can find a finite sequence {𝜉_𝑘}^𝑚𝑘=1 (𝑚 depending on 𝑟) in 𝕊_𝑁 with the following properties.(1)𝕊_𝑁=⋃_𝑘𝑄(𝜉_𝑘,𝑟).(2)The sets 𝑄(𝜉_𝑘,𝑟/4) are mutually disjoint.(3)Each point of 𝕊_𝑁 belongs to at most 𝑀 of the sets 𝑄(𝜉_𝑘,4𝑟).

From now on, 𝑀 denotes the constant involved in the previous lemma. Let 𝑛≥0 be an integer and let 𝐶_𝑛 be the corona𝐶_𝑛=𝑧∈𝔹_𝑁,1−12^𝑛≤|𝑧|<1−12^𝑛+1.(1.18) For any 𝑛≥0, let (𝜉_𝑛,𝑘)_𝑘⊂𝕊_𝑁 be given by Lemma 1.8 putting 𝑟=1/2^𝑛. For 𝑘≥0, we set𝑇_0,𝑘=𝑧∈𝔹_𝑁⧵{0},𝑧|𝑧|∈𝑄𝜉^0,𝑘,1∪{0}.(1.19) Then we define the sets 𝑇_𝑛,𝑘, for 𝑛≥1 and 𝑘≥0, by𝑇_𝑛,𝑘=𝑧∈𝔹_𝑁⧵{0},𝑧|𝑧|∈𝑄𝜉^𝑛,𝑘,12^𝑛.(1.20) We have both_𝑛≥0_𝑘≥0𝐶_𝑛=𝔹_𝑁,𝑇_0,𝑘=𝔹_𝑁,_𝑘≥0𝑇_𝑛,𝑘=𝔹_𝑁⧵{0},𝑛≥1.(1.21) For (𝑛,𝑘)∈ℕ², we finally define the subset Δ_(𝑛,𝑘) of 𝔹_𝑁 by Δ_(𝑛,𝑘)=𝐶_𝑛∩𝑇_𝑛,𝑘. These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function Λ_𝑓 for 𝑓∈𝐴^Ψ𝛼(𝔹_𝑁) byΛ_𝑓=_{𝑛,𝑘≥0}sup_{Δ(𝑛,𝑘)}||𝑓(𝑧)||𝜒_{Δ(𝑛,𝑘)},(1.22) where 𝜒_{Δ(𝑛,𝑘)} is the characteristic function of Δ_(𝑛,𝑘). Now we may easily adapt the proof of [11, Proposition 2.2] (which only relies on the subharmonicity of Ψ(|𝑓|)) to get the following.

Proposition 1.9. Let Ψ be a subharmonic-preserving growth function and let 𝛼>−1. Then the maximal operator Λ, which carries 𝑓 to Λ_𝑓, is bounded from 𝐴^Ψ𝛼 to 𝐿^Ψ(𝔹_𝑁,𝑣_𝛼). More precisely, there exists 𝐵≥1 such that for every 𝑓∈𝐴^Ψ𝛼, one has ‖‖Λ_𝑓‖‖_{𝐿Ψ(𝔹𝑁,𝑣𝛼)}≤𝐵‖𝑓‖_𝛼,Ψ.(1.23) In particular, a holomorphic function 𝑓 belongs to 𝐴^Ψ𝛼 if and only if Λ_𝑓 belongs to 𝐿^Ψ(𝔹_𝑁,𝑣_𝛼).

1.2.2. Large Hardy-Orlicz Spaces

Let Ψ still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let (Ω,ℙ)=(𝕊_𝑁,𝜎). The Hardy-Orlicz space 𝐻^Ψ of the ball consists of all holomorphic functions 𝑓 on 𝔹_𝑁 such that‖𝑓‖_𝐻^Ψ∶=sup_0<𝑟<1‖‖𝑓_𝑟‖‖_Ψ<∞,(1.24) where 𝑓_𝑟(𝑧)=𝑓(𝑟𝑧), and where ‖⋅‖_Ψ is the Luxembourg norm on the Orlicz space 𝐿^Ψ(𝕊_𝑁,𝜎). Note that we can replace sup_0<𝑟<1‖𝑓_𝑟‖_Ψ<∞ by lim_𝑟→1‖𝑓_𝑟‖_Ψ<∞ thanks to the subharmonicity of Ψ(|𝑓|). Because Ψ is supposed to be a growth function, we have the following inclusion:𝐻¹⊂𝐻^Ψ⊂𝐻^𝑝(1.25) for a growth function Ψ of order 𝑝. In particular, every 𝑓∈𝐻^Ψ admits a boundary radial limit, denoted by 𝑓^∗, 𝜎-almost everywhere on 𝕊_𝑁. Let us also note that if Ψ is a growth function, then 𝑓∈𝐻^Ψ if and only if‖𝑓‖^Ψ𝐻_Ψ∶=sup_0<𝑟<1‖‖𝑓_𝑟‖‖^Ψ=lim_𝑟→1‖‖𝑓_𝑟‖‖^Ψ<∞(Inequalities(1.7)and(1.8)).(1.26)

Without possible confusion, we will write ‖⋅‖_Ψ instead of ‖⋅‖_𝐻^Ψ (resp., ‖⋅‖^Ψ instead ‖⋅‖^Ψ𝐻_Ψ). As for Bergman-Orlicz spaces, a linear operator 𝑇 from 𝐻^Ψ1 to some 𝐿^Ψ2(𝕊_𝑁,𝜎) or 𝐻^Ψ2 where Ψ₁ and Ψ₂ are two growth functions is continuous (bounded) if and only if there exists a constant 𝐶>0 such thatmax‖𝑇(𝑓)‖_Ψ2,‖𝑇(𝑓)‖^Ψ2≤𝐶(1.27) for any 𝑓∈𝐻^Ψ1 such that min(‖𝑓‖_Ψ1,‖𝑓‖^Ψ1)≤1.

In addition, it is clear that, for any 𝛼>−1,𝐴^Ψ𝛼⊃𝐻^Ψ and ‖𝑓‖_𝛼,Ψ≤‖𝑓‖_Ψ for any 𝑓∈𝐻^Ψ. Therefore, letting 𝛼 tend to −1 in Proposition 1.6, we get the following.

Proposition 1.10. Let Ψ be a subharmonic-preserving growth function. For any 𝑎∈𝔹_𝑁 and any 𝑓∈𝐻^Ψ, one has ||𝑓(𝑎)||≤Ψ⁻¹1−1|𝑎|^𝑁‖𝑓‖^Ψ.(1.28)

As a corollary, we have the following.

Corollary 1.11. Let Ψ be a subharmonic-preserving growth function. 𝐻^Ψ is a complete metric space (with the equivalent distances induced by ‖⋅‖_Ψ and ‖⋅‖^Ψ, as usual).

For 𝑎∈𝔹_𝑁 and 𝑝>0, we introduce the “test’’ function 𝑓_𝑎,𝑝 defined for any 𝑧∈𝔹_𝑁 by𝑓_𝑎,𝑝(𝑧)=1−|𝑎|1−⟨𝑧,𝑎⟩^2𝑁/𝑝.(1.29) It is easily seen that 𝑓_𝑎,𝑝∈𝐻^∞ with ‖𝑓_𝑎,𝑝‖_∞=1 and that |𝑓_𝑎,𝑝(𝑧)|≤1 for any 𝑧∈𝕊_𝑁. Moreover, let us observe that||𝑓_𝑎,𝑝(𝑧)||^𝑝=1−|𝑎|1+|𝑎|^𝑁𝑃|𝑎|𝑧,𝑎|𝑎|,(1.30) so that ‖𝑓_𝑎,𝑝‖_𝑝=((1−|𝑎|)/(1+|𝑎|))^𝑁/𝑝. Therefore, if Ψ is a growth function of order 𝑝, we have, by Lemma 1.5,‖‖𝑓_𝑎,𝑝‖‖_Ψ≤1Ψ⁻¹1/‖‖𝑓^𝑎,𝑝‖‖_𝑝≤Ψ⁻¹(1/(11−|𝑎|))^𝑁/𝑝.(1.31)

It is very convenient to see 𝐻^Ψ as a closed subspace of 𝐿^Ψ(𝕊_𝑁). When Ψ is an Orlicz function, this is possible thanks to the representation of any function in 𝐻¹ by the Poisson integral of its boundary values. This does not work any more in 𝐻^𝑝 with 0<𝑝<1, even in this case. However, using a radial maximal function, we can still see 𝐻^𝑝 as a subspace of 𝐿^𝑝. We are going to extend this to 𝐻^Ψ for Ψ a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on 𝔹_𝑁: for (𝑧,𝑤)∈𝔹_𝑁,𝑑(𝑧,𝑤)=||1−⟨𝑧,𝑤⟩||.(1.32) It is well known that 𝑑 is a distance on 𝕊_𝑁 and a pseudodistance on 𝔹_𝑁 [21, Paragraph 5.1]. It permits to define the Korányi approach region Γ(𝜁) for 𝜁∈𝕊_𝑁:Γ(𝜁)=𝑧∈𝔹_𝑁,𝑑(𝑧,𝜁)²<1−|𝑧|².(1.33) Then the maximal function 𝑁_𝑓 of 𝑓, associated to Korányi approach region, is given by𝑁_𝑓(𝜁)=sup_{𝑧∈Γ(𝜁)}||𝑓(𝑧)||(1.34) for any 𝜁∈𝕊_𝑁. [17, Theorem 1.3] will be very useful.

Theorem 1.12. Let Ψ be a growth function. Then, for any 𝑓∈𝐻^Ψ, _𝕊𝑁Ψ||𝑁_𝑓||𝑑𝜎≲‖𝑓‖_Ψ.(1.35) In particular, a holomorphic function 𝑓 belongs to 𝐻^Ψ if and only if 𝑁_𝑓 belongs to 𝐿^Ψ(𝕊_𝑁,𝜎).

From this theorem, we deduce the following one.

Theorem 1.13. Let Ψ be a subharmonic-preserving growth function. Then for every 𝑓∈𝐻^Ψ, one has(1)lim_𝑟→1∫_𝕊𝑁Ψ(|𝑓^∗−𝑓_𝑟|)𝑑𝜎=0;(2)‖𝑓^∗‖^Ψ=‖𝑓‖^Ψ𝐻_Ψ;(3)𝐻^Ψ is separable. More precisely, the polynomials are dense in 𝐻^Ψ.

Proof. Let 𝑀_rad𝑓(𝜁)=sup_0<𝑟<1|𝑓(𝑟𝜁)| for 𝜁∈𝕊_𝑁. Obviously, 𝑀_rad𝑓≤𝑁_𝑓, hence ∫_𝕊𝑁Ψ(𝑀_rad𝑓)𝑑𝜎≲‖𝑓‖_Ψ (Theorem 1.12). Since Ψ is concave and vanishes at 0, we have Ψ(|𝑓^∗−𝑓_𝑟|)≤Ψ(|𝑓^∗|)+Ψ(|𝑓_𝑟|)≤2Ψ(𝑀_rad𝑓). Now Ψ(|𝑓^∗−𝑓_𝑟|(𝜁)) tends to 0 as 𝑟 goes to 1 for 𝜎-almost every 𝜁∈𝕊_𝑁. By the dominated convergence theorem, (1) follows.
Then ‖𝑓^∗‖^Ψ=lim_𝑟→1‖𝑓_𝑟‖^Ψ and (2) comes from to the subharmonicity of Ψ(|𝑓|).
We proved in (1) that 𝑓_𝑟 tends to 𝑓^∗ in 𝐻^Ψ for ‖⋅‖^Ψ (hence for ‖⋅‖_Ψ also). We approach every 𝑓_𝑟 uniformly on 𝔹_𝑁 by its Taylor series to get the third assertion.

2. Carleson Embedding Theorems

2.1. Statements of the Results

For 𝜁∈𝔹_𝑁 and ℎ∈]0,1], we define the nonisotropic “ball’’ of 𝔹_𝑁 by𝑆(𝜁,ℎ)=𝑧∈𝔹_𝑁,𝑑(𝜁,𝑧)²<ℎ(2.1) and its analogue in 𝔹_𝑁 by𝒮(𝜁,ℎ)=𝑧∈𝔹_𝑁,𝑑(𝜁,𝑧)²<ℎ.(2.2) Let us also denote by𝑄(𝜁,ℎ)=𝒮(𝜁,ℎ)∩𝕊_𝑁,(2.3) the “true’’ balls in 𝕊_𝑁. We have 𝜎(𝑄(𝜁,ℎ))≈ℎ^𝑁 and 𝑣_𝛼(𝑆(𝜁,ℎ))≈ℎ^𝑁+𝛼+1 [22].

Let 𝜇 be a positive Borel measure on 𝔹_𝑁 whose restriction to 𝕊_𝑁 is absolutely continuous with respect to 𝜎 and let 0<𝑝≤𝑞<∞. By definition, 𝜇 is a (𝑞/𝑝)-Carleson measure if 𝜇(𝒮(𝜁,ℎ))≤𝐶ℎ^{𝑁𝑞/𝑝}, while it is a vanishing (𝑞/𝑝)-Carleson measure if 𝜇(𝒮(𝜁,ℎ))=𝑜(ℎ^{𝑁𝑞/𝑝}) when ℎ goes to 0. A variant of the well-known Carleson theorem for Hardy spaces [1, 3] ensures that the embedding 𝐻^𝑝↪𝐿^𝑞(𝔹_𝑁,𝜇) is bounded (resp., compact) if and only if 𝜇 is a (𝑞/𝑝)-Carleson measure (resp., a vanishing (𝑞/𝑝)-Carleson measure).

Similarly, we define the (𝛼,(𝑞/𝑝))-Bergman Carleson measures (resp., vanishing (𝛼,(𝑞/𝑝))-Bergman Carleson measures) for weighted Bergman spaces by 𝜇(𝑆(𝜁,ℎ))≤𝐶ℎ^{(𝑁+𝛼+1)𝑞/𝑝} (resp., 𝜇(𝑆(𝜁,ℎ))=𝑜(ℎ^{(𝑁+𝛼+1)𝑞/𝑝})). When 𝑝=𝑞, we just speak about 𝛼-Bergman Carleson measures (resp., vanishing 𝛼-Bergman Carleson measures). Ueki [7] showed that 𝐴^𝑝𝛼↪𝐿^𝑞(𝔹_𝑁,𝜇) is bounded (resp., compact) if and only if 𝜇 is a (𝛼,(𝑞/𝑝))-Bergman Carleson measure (resp., a vanishing (𝛼,(𝑞/𝑝))-Carleson measure).

In the context of Hardy-Orlicz spaces (resp., weighted Bergman-Orlicz spaces) smaller than 𝐻¹ (resp., 𝐴^1𝛼) (i.e., when the defining function Ψ 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 𝐻¹ (resp., 𝐴^1𝛼), we state that the characterizations of the boundedness and compactness of 𝐻^Ψ1↪𝐿^Ψ2(𝔹_𝑁,𝜇) (resp., 𝐴^Ψ1_𝛼↪𝐿^Ψ2(𝔹_𝑁,𝜇)), where Ψ₁ and Ψ₂ are two growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing at least for large values of 𝑥 (or equivalently 𝑥↦Ψ₂∘Ψ¹⁻¹(𝑥)/𝑥 nondecreasing, since Ψ₁ is increasing and does vanish except in 0), only depend on the growth of Ψ₂∘Ψ¹⁻¹ at infinity.

Note that if Ψ_𝑖(𝑥)=𝑥^𝑝𝑖, Ψ_𝑖=Φ^𝑝𝑖 (with Φ an Orlicz function), or Ψ_𝑖(𝑥)=𝑥^𝑝𝑖log^𝑞𝑖(𝐶+𝑥) with 𝑝₂≥𝑝₁, then 𝑥↦Ψ₂∘Ψ¹⁻¹(𝑥)/𝑥 is nondecreasing.

Theorem 2.1. Let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing. Let 𝜇 be a finite positive Borel measure on 𝔹_𝑁 (whose restriction to 𝕊_𝑁 is absolutely continuous with respect to 𝜎). Then,(1)𝐻^Ψ1 embeds into 𝐿^Ψ2(𝔹_𝑁,𝜇) if and only if there exists some ℎ₀∈(0,1) such that, for any ℎ∈(0,ℎ₀), 𝜇(𝒮(𝜁,ℎ))≲1Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁(2.4) uniformly in 𝜁∈𝕊_𝑁,(2)the embedding 𝐻^Ψ1↪𝐿^Ψ2(𝔹_𝑁,𝜇) is compact if and only if 𝜇(𝒮(𝜁,ℎ))=𝑜_ℎ→0Ψ₂∘Ψ1¹⁻¹1/ℎ^𝑁(2.5) uniformly in 𝜁∈𝕊_𝑁.A measure 𝜇 which satisfies (2.4) (resp., (2.5)) will be called a (Ψ₁,Ψ₂)-Carleson measure (resp., a vanishing (Ψ₁,Ψ₂)-Carleson measure).

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

Theorem 2.2. Let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing, and let 𝛼>−1. Let also 𝜇 be a finite positive Borel measure on 𝔹_𝑁. Then,(1)𝐴^Ψ1_𝛼 embeds into 𝐿^Ψ2(𝔹_𝑁,𝜇) if and only if there exists ℎ₀∈(0,1) such that, for any ℎ∈(0,ℎ₀), 𝜇(𝑆(𝜁,ℎ))≲1Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁+𝛼+1,(2.6) uniformly in 𝜁∈𝕊_𝑁,(2)the embedding 𝐴^Ψ1_𝛼↪𝐿^Ψ2(𝔹_𝑁,𝜇) is compact if and only if 𝜇(𝑆(𝜁,ℎ))≤𝑜_ℎ→0Ψ₂∘Ψ¹⁻¹11/ℎ^𝑁+𝛼+1,(2.7) uniformly in 𝜁∈𝕊_𝑁.A measure 𝜇 which satisfies (2.6) (resp., (2.7)) will be called a (𝛼,Ψ₁,Ψ₂)-Bergman-Carleson measure (resp., a vanishing (𝛼,Ψ₁,Ψ₂)-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 𝛼>−1 and let 0<𝑝≤𝑞<1 and Ψ₁=Φ^𝑝, Ψ₂=Φ^𝑞 as in (3) of Example 1.3. Let 𝜇 be a finite positive Borel measure on 𝔹_𝑁 (whose restriction to 𝕊_𝑁 is absolutely continuous with respect to 𝜎) (resp., on 𝔹_𝑁). Then,(1)𝐻^Φ𝑝 (resp., 𝐴^Φ𝑝_𝛼) embeds into 𝐿^Φ𝑞(𝔹_𝑁,𝜇) (resp., 𝐿^Φ𝑞(𝔹_𝑁,𝜇)) if and only if 𝜇 is a (𝑞/𝑝)-Carleson measure (resp., a (𝛼,(𝑞/𝑝))-Bergman-Carleson measure),(2)the embedding 𝐻^Φ𝑝↪𝐿^Φ𝑞(𝔹_𝑁,𝜇) (resp., 𝐴^Φ𝑝_𝛼↪𝐿^Φ𝑞(𝔹_𝑁,𝜇)) is compact if and only if 𝜇 is a vanishing (𝑞/𝑝)-Carleson measure (resp., a vanishing (𝛼,(𝑞/𝑝)) Bergman-Carleson measure).

If Ψ₁=Ψ₂ (or equivalently if Ψ₁ and Ψ₂ are equivalent), we have the following.

Corollary 2.5. Let 𝛼>−1, and let Ψ be subharmonic-preserving growth function. Let 𝜇 be a finite positive Borel measure on 𝔹_𝑁 (whose restriction to 𝕊_𝑁 is absolutely continuous with respect to 𝜎) (resp., on 𝔹_𝑁). Then,(1)𝐻^Ψ (resp., 𝐴^Ψ𝛼) embeds into 𝐿^Ψ(𝔹_𝑁,𝜇) (resp., 𝐿^Ψ(𝔹_𝑁,𝜇)) if and only if 𝜇 is a Carleson measure (resp., a 𝛼-Bergman-Carleson measure),(2)the embedding 𝐻^Ψ↪𝐿^Ψ(𝔹_𝑁,𝜇) (resp., 𝐴^Ψ𝛼↪𝐿^Ψ(𝔹_𝑁,𝜇)) is compact if and only if 𝜇 is a vanishing Carleson measure (resp., a vanishing 𝛼-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 𝐻^Ψ (resp., 𝐴^Ψ𝛼), for Ψ a subharmonic-preserving growth function, implies the convergence on every compact subset of 𝔹_𝑁, but it stems from Proposition 1.10 (resp., Proposition 1.6).

Proposition 2.6. Let 𝛼>−1, let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions and let 𝜇 (resp., 𝜇) be a finite positive Borel measure on 𝔹_𝑁 (resp., 𝔹_𝑁) whose restriction to 𝕊_𝑁 is absolutely continuous with respect to 𝜎. One assume that 𝑗_𝜇∶𝐻^Ψ1↪𝐿^Ψ2(𝔹_𝑁,𝜇) (resp., 𝑗_𝜇𝛼∶𝐴^Ψ1_𝛼↪𝐿^Ψ2(𝔹_𝑁,𝜇)) is well defined (hence bounded).(1)The two following assertions are equivalent:(a)the canonical embedding 𝑗_𝜇 (resp., 𝑗_𝜇𝛼) is compact;(b)every sequence in the unit ball of 𝐻^Ψ1 (resp., 𝐴^Ψ1_𝛼), which is convergent to 0 uniformly on every compact subset of 𝔹_𝑁, is convergent to 0 in 𝐿^Ψ2(𝜇) (resp., 𝐿^Ψ2(𝜇)).(2)If lim_𝑟→1⁻‖𝐼_𝑟‖=0 (resp., lim_𝑟→1⁻‖𝐼_𝑟,𝛼‖=0), where 𝐼_𝑟(𝑓)=𝑓⋅𝜒_{𝔹𝑁⧵𝑟𝔹𝑁} (resp., 𝐼_𝑟(𝑓)=𝑓⋅𝜒_{𝔹𝑁⧵𝑟𝔹𝑁}), then the canonical embedding 𝑗_𝜇 (resp., 𝑗_𝜇𝛼) 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 𝐾_𝜇, associated to 𝜇 by𝐾_𝜇(ℎ)=sup_0<𝑡<ℎsup_{𝜁∈𝕊𝑁}𝜇(𝒮(𝜁,𝑡))𝑡^𝑁,forℎ∈(0,1),(2.8) so that 𝜇 is a Carleson measure if and only if 𝐾_𝜇(ℎ) is finite for some ℎ∈(0,1).

Theorem 2.7 (see [12, Theorem 2.4]). There exist two constants 𝐶>0 and 𝐶>1 such that, for every 𝑓 continuous on 𝔹_𝑁 which admits boundary values almost everywhere on 𝕊_𝑁, and every finite positive Borel measure 𝜇 on 𝔹_𝑁, one has 𝜇𝑧∈𝔹_𝑁,|𝑧|>1−ℎ,||𝑓(𝑧)||>𝑡≤𝐶𝐾_𝜇(𝐶ℎ)𝜎𝑁_𝑓>𝑡,(2.9) for every ℎ∈(0,1/𝐶) and every 𝑡>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 𝜇 be a finite positive Borel measure on 𝔹_𝑁 and let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions. Let 𝐶≥1 be the constant appearing in Theorem 2.7. Assume that there exist 𝐴>0,𝜂>0 and ℎ_𝐴∈(0,1/𝐶) such that 𝐾_𝜇(ℎ)≤𝜂1/ℎ^𝑁Ψ₂𝐴Ψ¹⁻¹1/ℎ^𝑁,(2.10) for every ℎ∈(0,ℎ_𝐴). Then, for every 𝑓∈𝐻^Ψ1(𝔹_𝑁) such that ‖𝑓‖_Ψ1≤1 and every Borel subset 𝐸 of 𝔹_𝑁, _𝐸Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+𝜂_𝕊𝑁Ψ₁𝑁_𝑓𝑑𝜎,(2.11) where ≲ involves a constant which is independent of 𝑓, 𝜂, and 𝐸.

Proof. Let 𝑓∈𝐻^Ψ1(𝔹_𝑁) with ‖𝑓‖_Ψ1≤1. With the notations of the statement of the lemma and using Proposition 1.10, the proof of [12, Lemma 2.6] directly yields. _𝐸Ψ₂2^𝑁+1(𝐶+𝐴1)𝐶^𝑁−1||𝑓||𝑑𝜇≤+^𝑥𝐴₀𝜂𝐶Ψ^′₂(𝑠)𝜇(𝐸)𝑑𝑠𝐶^𝑁^∞𝑥_𝐴Ψ^2(𝑠)Ψ¹(𝐶+1)𝐶^𝑁−1/𝐴𝑆×𝜎𝑁_𝑓Ψ₂(((𝐶+1)/𝐶)𝑆)>2^𝑁+1(𝐶+1)𝐶^𝑁−1𝐴𝑠𝑑𝑠,(2.12) for some constant 𝑥_𝐴 which depends only on 𝐴. Now, since Ψ₁ and Ψ₂ are two concave growth functions, we have Ψ_𝑖(𝑠)≲Ψ_𝑖(𝐾𝑠)≲Ψ_𝑖(𝑠),(2.13) where ≲ involve constants which only depend on 𝐾 and Ψ_𝑖. Therefore, (2.12) becomes _𝐸Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+𝜂^∞𝑥_𝐴Ψ^2(𝑠)Ψ¹(𝑠)Ψ₂(𝑠)𝜎𝑁^𝑓>2^𝑁+1(𝐶+1)𝐶^𝑁−1𝐴𝑠𝑑𝑠.(2.14) Since 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing and since Ψ₁ and Ψ₂ and their derivatives do not vanish except in 0, we have, for any 𝑠>0, Ψ^1(𝑠)Ψ^2(𝑠)≤Ψ¹(𝑠)Ψ₂(𝑠).(2.15) It follows that _𝐸Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+𝜂^∞𝑥_𝐴Ψ^1(𝑠)𝜎𝑁_𝑓>2^𝑁+1(𝐶+1)𝐶^𝑁−1≲𝜇(𝐸)+𝜂_0+∞𝐴𝑠𝑑𝑠Ψ₁2^𝑁+1(𝐶𝐴+𝑁1^𝑓)𝐶^𝑁−1=𝜇(𝐸)+𝜂_𝕊𝑁>𝑠𝑑𝑠𝜓₁≲𝜇(𝐸)+𝜂_𝕊𝑁2^𝑁+1(𝐶𝐴+𝑁1^𝑓)𝐶^𝑁−1𝑑𝜎𝜓₁𝑁_𝑓𝑑𝜎.(2.16)

We now can prove Theorem 2.1.

Proof of Theorem 2.1. (1) We assume that 𝜇 is a (Ψ₁,Ψ₂)-Carleson measure, and we intend to show that 𝑗_𝜇∶𝐻^Ψ1↪𝐿^Ψ2(𝔹_𝑁,𝜇) is well defined (hence bounded). We observe that (2.10) is satisfied for 𝐴=𝜂=1. Indeed we have, for any ℎ∈(0,ℎ₀), 𝜇(𝒮(𝜁,ℎ))ℎ^𝑁≲1/ℎ^𝑁Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁,(2.17) uniformly in 𝜁∈𝕊_𝑁. Now, 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing (at least for large 𝑥) so that we can find ℎ₁∈(0,min(1/𝐶,ℎ₀)) such that, for any ℎ∈(0,ℎ₁), 𝐾_𝜇(ℎ)≲1/ℎ^𝑁Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁.(2.18) Therefore, we may and will apply Lemma 2.8 with 𝐸=𝔹_𝑁 to get _𝔹𝑁Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+_𝕊NΨ₁𝑁_𝑓𝑑𝜎.(2.19) We finish the proof of (1) using Theorem 1.12 and Inequality (1.7): _𝔹𝑁Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+‖𝑓‖_Ψ1.(2.20)
For the converse, let 𝑔_𝑎,𝑝=Φ⁻¹((1/(1−|𝑎|))^𝑁/𝑝)𝑓_𝑎,𝑝, where 𝑓_𝑎,𝑝 is the test function introduced at the end of Section 1.2.2, with 𝑝 such that Ψ₁ is a growth function of order 𝑝. According to (1.31), 𝑔_𝑎,𝑝 lies in the unit ball of 𝐻^Ψ1. Let 𝐶 be the (finite) norm of 𝑗_𝜇. Using a classical computation which gives that |1−⟨𝑧,𝑎⟩|≤2(1−|𝑎|) for any 𝑧∈𝒮(𝑎/|𝑎|,1−|𝑎|), we get |𝑓_𝑎,𝑝(𝑧)|≥1/4^𝑁/𝑝 for such 𝑧, hence 1≥_𝔹𝑁Ψ₂||𝑔^𝑎,𝑝||𝐶𝑑𝜇≥𝜇𝒮|𝑎𝑎|,1−|𝑎|Ψ²41^𝑁/𝑝𝐶Ψ¹⁻¹1−1|𝑎|^𝑁/𝑝(2.21) for any 𝑎∈𝔹_𝑁. We may assume that 𝐶≥1 and since Ψ₂ is concave and vanishes at 0, it follows that 𝜇𝒮𝑎|𝑎|,1−|𝑎|≤4^𝑁/𝑝𝐶1Ψ₂∘Ψ¹⁻¹1/(1−|𝑎|)^𝑁.(2.22)
(2) We turn to the compactness part. We first prove the sufficient part. Let us assume that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing and that 𝜇 is a vanishing (Ψ₁,Ψ₂)-Carleson measure. In particular, 𝜇 is a (Ψ₁,Ψ₂)-Carleson measure and we may apply Proposition 2.6. Then it is sufficient to prove that for every 𝜀>0, there exists 𝑟 close enough to 1, such that ‖𝐼_𝑟‖<𝜀 (where 𝐼_𝑟∶𝐻^Ψ1→𝐿^Ψ2(𝔹_𝑁⧵𝑟𝔹_𝑁,𝜇)). We fix 𝜀>0 and let 𝑓 be in the unit ball of 𝐻^Ψ2. As in the proof of the sufficient part of (1), since 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing, 𝜇 is a vanishing (Ψ₁,Ψ₂)-Carleson measure implying that 𝐾_𝜇(ℎ)≲𝜀1/ℎ^𝑁Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁,(2.23) for ℎ small enough. Then, we apply Lemma 2.8 with 𝜂=𝜀 and 𝐸=𝔹_𝑁⧵𝑟𝔹_𝑁 to get the existence of 𝑟 close enough to 1 (independent of 𝑓), such that ‖‖𝐼_𝑟(𝑓)‖‖_Ψ2≲≲𝜇𝜇𝔹_𝑁⧵𝑟𝔹_𝑁+𝜀_𝕊𝑁Ψ₁𝑁_𝑓𝑑𝜎𝔹_𝑁⧵𝑟𝔹_𝑁+𝜀‖𝑓‖^Ψ₁,(2.24) for any 𝜀>0. Now, we may argue as in [12, Lemma 2.13] to prove that, under the assumption that 𝜇 is a (Ψ₁,Ψ₂)-Carleson measure, 𝜇(𝕊_𝑁)=0. Hence limsup_𝑟→1‖‖𝐼_𝑟(𝑓)‖‖_Ψ2≲𝜀‖𝑓‖_Ψ1,(2.25) for any 𝜀>0, which gives (2).
For the converse, we assume that 𝑗_𝜇 is compact but that 𝜇 is not a vanishing (Ψ₁,Ψ₂)-Carleson measure, that is there exist 𝜀₀>0, a sequence (ℎ_𝑛)_𝑛 decreasing to 0 and a sequence (𝜁_𝑛)_𝑛⊂𝕊_𝑁 such that 𝜇𝒮𝜁_𝑛,ℎ_𝑛≥𝜀₀1Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁𝑛.(2.26) Now, let 𝑔_{𝑎𝑛,𝑝}=Φ⁻¹((1/(1−|𝑎_𝑛|))^𝑁/𝑝)𝑓_{𝑎𝑛,𝑝} be as in the proof of the necessity in the boundedness part with 𝑎_𝑛=(1−ℎ_𝑛)𝜁_𝑛 and 𝑝 such that Ψ₁ is growth function of order 𝑝. Since (𝑔_{𝑎𝑛,𝑝})_𝑛 is bounded in 𝐻^Ψ1 and converges uniformly on every compact subset of 𝔹_𝑁, then we must have ‖𝑗_𝜇(𝑔_{𝑎𝑛,𝑝})‖_Ψ2→_𝑛→∞0, because of Proposition 2.6. But, for 𝑧∈𝒮(𝜁_𝑛,ℎ_𝑛), we saw that ||𝑔_{𝑎𝑛,𝑝}(𝑧)||≥14^𝑁/𝑝Ψ¹⁻¹ℎ1^𝑁𝑛.(2.27) Since Ψ₂ is a growth function of order, let say 𝑞, we have, for any 𝑡∈(0,1), Ψ₂(𝑦/𝑡)≥Ψ₂(𝑦)/𝑡^𝑞. Hence, we get _𝔹𝑁Ψ₂4^{𝑁𝑞/𝑝}||𝑔_{𝑎𝑛,𝑝}||𝜀^01/𝑞𝑑𝜇≥Ψ²𝜀1^01/𝑞≥ΨΨ¹⁻¹1/ℎ^𝑁𝑛𝜇𝒮𝜁_𝑛,ℎ_𝑛₂∘Ψ¹⁻¹1/ℎ^𝑁𝑛𝜀₀𝜀₀1=1.Ψ₂∘Ψ¹⁻¹1/ℎ^𝑁𝑛(2.28) Therefore, ‖𝑗_𝜇(𝑔_{𝑎𝑛,𝑝})‖_Ψ2≥𝜀^01/𝑞/4^{𝑁𝑞/𝑝}, which is a contradiction.

2.2.2. Proof of Theorem 2.2

We introduce the following function:𝐾_𝜇,𝛼(ℎ)=sup_0<𝑡<ℎsup_{𝜁∈𝕊𝑁}𝜇(𝑆(𝜁,𝑡))𝑡^𝑁+𝛼+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 𝐶>0 such that, for every 𝑓 continuous on 𝔹_𝑁 and every positive finite Borel measure 𝜇 on 𝔹_𝑁, one has 𝜇𝑧∈𝔹_𝑁,|𝑧|>1−ℎ,||𝑓(𝑧)||>𝑡≤𝐶𝐾_𝜇,𝛼(2ℎ)𝑣_𝛼Λ_𝑓>𝑡,(2.30) for every ℎ∈(0,1/2) and every 𝑡>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 𝜇 be a finite positive Borel measure on 𝔹_𝑁 and let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions. Assume that there exist 𝐴>0, 𝜂>0, and ℎ_𝐴∈(0,1/2) such that 𝐾_𝜇,𝛼(ℎ)≤𝜂1/ℎ^𝑁+1+𝛼𝜓₂𝐴Ψ¹⁻¹1/ℎ^𝑁+1+𝛼,(2.31) for every ℎ∈(0,ℎ_𝐴). Then, for every 𝑓∈𝐴^Ψ1_𝛼(𝔹_𝑁) with ‖𝑓‖_𝐴^Ψ1𝛼≤1 and every Borel subset 𝐸 of 𝔹_𝑁, one has _𝐸Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+𝜂_𝔹𝑁Ψ₁Λ_𝑓𝑑𝑣_𝛼,(2.32) where ≲ involves constants which are independent of 𝑓, 𝜂, and 𝐸.

We now prove Theorem 2.2.

Proof of Theorem 2.2. Let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing.
(1) We first assume that 𝜇 is a (𝛼,Ψ₁,Ψ₂)-Bergman Carleson measure, which implies, as in the beginning of the proof of (1) of Theorem 2.1, that condition (2.31) with 𝐴=𝜂=1 is fulfilled. Applying (2.31) to 𝑓, with 𝐸=𝔹_𝑁, gives _𝔹𝑁Ψ₂||𝑓||𝑑𝜇≲𝜇(𝐸)+_𝔹𝑁Ψ₁Λ_𝑓𝑑𝑣_𝛼.(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 𝑔_{𝑎,𝛼,𝑝}=Φ⁻¹((1/(1−|𝑎|))^{(𝑁+𝛼+1)/𝑝})𝑓_{𝑎,𝛼,𝑝}, where 𝑓_{𝑎,𝛼,𝑝} has been introduced in Section 1.2.1, with 𝑝 such that Ψ₁ is a growth function of order 𝑝.
(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 𝑓∈𝐴^Ψ1_𝛼, with 𝜂=𝜀 and 𝐸=𝔹_𝑁⧵𝑟𝔹_𝑁 to show that ‖𝑓.𝜒_{𝔹𝑁⧵𝑟𝔹𝑁}‖_Ψ2 tends to 0, as 𝑟 tends to 1. Proposition 2.6 then shows that 𝐴^Ψ1_𝛼↪𝐿^Ψ2(𝜇) 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 𝐻^Ψ and 𝐴^Ψ𝛼. As usual, for 𝜙∶𝔹_𝑁→𝔹_𝑁 holomorphic, 𝐶_𝜙 will be seen as an embedding operator. To do so, we define the pullback measures 𝜇_𝜙 and 𝜇_𝜙,𝛼 of, respectively, 𝜎 and 𝑣_𝛼 under 𝜙:𝜇_𝜙(𝐸)=𝜎𝜙^∗−1(𝐸)∩𝕊_𝑁,(3.1) where 𝜙^∗ is the radial limit almost everywhere of 𝜙 and 𝐸 is any Borel subset of 𝔹_𝑁. And𝜇_𝜙,𝛼(𝐸)=𝑣_𝛼𝜙⁻¹(𝐸)(3.2) for any Borel subset 𝐸 of 𝔹_𝑁.

By a classical formula for pull-back measures,‖‖𝑗_𝜇𝜙(𝑓)‖‖_Ψ=‖‖𝐶_𝜙(𝑓)‖‖_Ψresp.,‖‖𝑗^𝜇_𝜙,𝛼(𝑓)‖‖_𝛼,Ψ=‖‖𝐶_𝜙(𝑓)‖‖_𝛼,Ψ(3.3) for any growth function Ψ and every polynomial 𝑓 (resp., for any 𝑓∈𝐴^Ψ𝛼). Then we extend this equality to the whole space 𝐻^Ψ by density of polynomials in 𝐻^Ψ (Theorem 1.13) and using Cauchy’s formula.

Remark 3.1. Observe that it is not true that 𝑗_𝜇𝜙(𝑓)=𝐶_𝜙(𝑓) for any 𝑓∈𝐻^Ψ but it is for any polynomial 𝑓. Such an equality holds for every function in 𝐴^Ψ𝛼.

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

Proposition 3.2. Let 𝛼>−1 and let Ψ₁ and Ψ₂ be as in Proposition 2.6. Let also 𝜙∶𝔹_𝑁→𝔹_𝑁 be holomorphic. 𝐶_𝜙 is compact from 𝐻^Ψ1 into 𝐻^Ψ2 (resp., 𝐴^Ψ1_𝛼 into 𝐴^Ψ2_𝛼) if and only if, for every bounded sequence (𝑓_𝑛)_𝑛⊂𝐻^Ψ1 (resp., (𝑓_𝑛)_𝑛⊂𝐴^Ψ1_𝛼) which converges to 0 on every compact of 𝔹_𝑁,‖𝐶_𝜙(𝑓)‖_Ψ2 (resp., ‖𝐶_𝜙(𝑓)‖_Ψ2,𝛼) 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 𝐶_𝜙 on 𝐻^Ψ and 𝐴^Ψ𝛼.

Theorem 3.3. Let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing. Let also 𝜙∶𝔹_𝑁→𝔹_𝑁 be holomorphic. Then,(1)𝐶_𝜙 is bounded from 𝐻^Ψ1 into 𝐻^Ψ2 if and only if 𝜇_𝜙 is a (Ψ₁,Ψ₂)-Carleson measure,(2)𝐶_𝜙 is compact from 𝐻^Ψ1 into 𝐻^Ψ2 if and only if 𝜇_𝜙 is a vanishing (Ψ₁,Ψ₂)-Carleson measure.

For Bergman-Orlicz spaces, we have the following.

Theorem 3.4. Let 𝛼>−1 and let Ψ₁ and Ψ₂ be two subharmonic-preserving growth functions such that 𝑥↦Ψ₂(𝑥)/Ψ₁(𝑥) is nondecreasing. Let also 𝜙∶𝔹_𝑁→𝔹_𝑁 be holomorphic. Then,(1)𝐶_𝜙 is bounded from 𝐴^Ψ1_𝛼 into 𝐴^Ψ2_𝛼 if and only if 𝜇_𝜙,𝛼 is a (𝛼,Ψ₁,Ψ₂)-Bergman Carleson measure,(2)𝐶_𝜙 is compact from 𝐴^Ψ1_𝛼 into 𝐴^Ψ2_𝛼 if and only if 𝜇_𝜙,𝛼 is a vanishing (𝛼,Ψ₁,Ψ₂)-Bergman Carleson measure.

We immediately deduce the following corollary, which reminds a similar result when the function Ψ is an Orlicz function satisfying the Δ₂-Condition (see [11, 12]).

Corollary 3.5. Let 𝛼>−1 and let Ψ be a subharmonic preserving growth function. Let also 𝜙∶𝔹_𝑁→𝔹_𝑁 be holomorphic. Then,(1)C_𝜙 is bounded (resp., compact) on 𝐻^Ψ if and only if it is bounded (resp., compact) on one (or equivalently every) 𝐻^𝑝, if and only if 𝜇_𝜙 is a Carleson measure (resp., a vanishing Carleson measure),(2)𝐶_𝜙 is bounded (resp., compact) on 𝐴^Ψ𝛼 if and only if it is bounded (resp., compact) on one (or equivalently every) 𝐴^𝑝𝛼, if and only if 𝜇_𝜙 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 𝛼>−1, let 0<𝑝≤𝑞<∞ and Ψ₁=Φ^𝑝, and Ψ₂=Φ^𝑞 as in (3) of Example 1.3. Let also 𝜙∶𝔹_𝑁→𝔹_𝑁 be holomorphic. Then,(1)𝐶_𝜙 is bounded (resp., compact) from 𝐻^Ψ1 to 𝐻^Ψ2 if and only if it is bounded (resp., compact) from 𝐻^𝑝 to 𝐻^𝑞, if and only if 𝜇_𝜙 is a 𝑞/𝑝-Carleson measure (resp., a vanishing 𝑞/𝑝-Carleson measure),(2)𝐶_𝜙 is bounded (resp., compact) from 𝐴^Ψ1_𝛼 to 𝐴^Ψ2_𝛼 if and only if it is bounded (resp., compact) from 𝐴^𝑝𝛼 to 𝐴^𝑞𝛼, if and only if 𝜇_𝜙 is a 𝑞/𝑝-Bergman-Carleson measure (resp., a vanishing 𝑞/𝑝-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.