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

Authors: Stéphane Charpentier; Benoît Sehba
Journal: Journal of Function Spaces and Applications
 (2012)

---

## Abstract

We characterize those measuresμ for which the Hardy-Orlicz (resp., weighted Bergman-Orlicz) space HΨ1 (resp., AαΨ1) of the unit ball of ℂN embeds boundedly or compactly into the Orlicz space LΨ2(𝔹N¯,μ) (resp., LΨ2(𝔹N,μ)), when the defining functions Ψ1 and Ψ2 are growth functions such that L1⊂LΨj 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 HΨ1 (resp., AαΨ1) into HΨ2 (resp., AαΨ2).


---

## Body

## 1. Introduction and Preliminaries

### 1.1. Introduction

Let𝔹N={z=(z1,…,zN)∈ℂN:|z|2=∑i=1N|zi|2<1} and 𝕊N=∂𝔹N, N≥1, denoting, respectively, the unit ball and the unit sphere of ℂN. For N=1, we denote by 𝔻 the unit disc of the complex plane.For a large class of spacesX of holomorphic functions in the unit disc or the unit ball, characterizations of the boundedness and compactness of the canonical embedding X↪Lp(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, whenX=Hp(𝔻), Carleson [1] proved that Hp(𝔻)↪Lp(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 Hp(𝔻)↪Lq(μ), with 0<p≤q<∞, in terms of (q/p)-Carleson measures. For the unweighted and weighted Bergman spaces Aαp(𝔹N), N≥1, similar results were obtained by Cima and Wogen [5], Luecking [6], Ueki [7]. We recall also that the compactness of X↪Lp(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 𝔹N¯, then μ∣𝕊N is absolutely continuous with respect to 𝕊N. 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 ofHp(𝔹N)↪Lq(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 𝕊N 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 Hp and H∞.This motivation is reinforced by a second observation: if the measureμϕ is the pull-back measure of the invariant-rotation measure on 𝕊N under a holomorphic map ϕ:𝔹N→𝔹N, N≥1, then μϕ 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 Hp and H∞ for any N≥1. This observation is directly connected to the study of composition operators Cϕ on Hp, which are defined by Cϕ(f)=f∘ϕ for f∈H(𝔹N), and which may be seen as the embedding operators Hp↪Lp(μϕ). By the way, this leads the authors of [9, 10] to state Carleson theorems for Hardy-Orlicz and Bergman-Orlicz spaces (resp., denoted by HΨ(𝔻) and AΨ(𝔻)) in the unit disc, when the defining function Ψ is an Orlicz function. These spaces appear as good candidates for generalizing Hp and Ap 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=μϕ is the pull-back measure under a holomorphic self-map ϕ of 𝔹N, then HΨ always embeds into LΨ(μ) whenever Ψ satisfied a fast growth condition (namely, the Δ2-Condition, that implies Ψ(x)≥ex for large value of x and means that we are close to H∞). Similarly, it was shown in [13] that if HΨ compactly embeds into LΨ(μ) for everyΨ, then H∞↪L∞(μ) compactly (the converse easily holds). We mention that there is no Ψ such that HΨ↪LΨ(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 HΨ↪LΨ(μ) boundedly (resp., compactly) if and only if Hp↪Lp(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 thanH1(𝔹N). 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 HΨ(𝔹N)↪Lq(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 AαΨ(𝔻), where Ψ=log+p, 1≤p<∞, that is, when AαΨ(𝔻) is the area Nevanlinna space. These measures reveal to be those μ for which Aαp↪Lp(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 thegood 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 HΨ1(𝔹N)↪LΨ2(𝔹N,μ) (resp., AαΨ1(𝔹N)↪LΨ2(𝔹N,μ)) is bounded or compact, when Ψ1 and Ψ2 are two growth functions (i.e., for which H1⊂HΨi⊂Hp (resp., Aα1⊂AαΨi⊂Aαp) 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 Aα1, 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 Aα1 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∈ℂN, the euclidean inner product of z and w will be denoted by 〈z,w〉, that is, 〈z,w〉=∑i=1Nziwi¯; 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,vα will be the measure on 𝔹N defined by dvα=cα(1-|z|)αdv, where v is the Lebesgue measure on 𝔹N and cα 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 writeHψ (resp., Aαψ) instead of Hψ(𝔹N) (resp., Aαψ(𝔹N)).


### 1.2. Preliminaries: Menagerie of Spaces

LetΨ:[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 LΨ(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 MΨ(Ω,ℙ) by(1.1)MΨ(Ω,P)={f:Ω⟶Cmeasurable;∫ΩΨ(|f|C)<∞foranyC>0},
and we also introduce the following set:(1.2)LΨ(Ω,P)={f:Ω⟶Cmeasurable;∫ΩΨ(|f|)<∞}.
In general, these three sets are not vector spaces and do not coincide, but we trivially have(1.3)MΨ(Ω,P)⊂LΨ(Ω,P)⊂LΨ(Ω,P).
We also define the Luxembourg gauge on LΨ(Ω,ℙ) by(1.4)‖f‖Ψ=inf⁡(λ>0,∫ΩΨ(|f|λ)≤1).
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 functionsΨ1 and Ψ2 as above are equivalent if there exists some constant c such that(1.5)cΨ1(cx)≤Ψ2(x)≤c-1Ψ1(c-1x),
for any x large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.In order to define a good topology onLΨ(Ω,ℙ) and to get properties convenient for our purpose, we will assume that Ψ satisfies the following definition.Definition 1.1.
Let0<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)≤ypΨ(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 thatψ is a growth function if it is a growth function of order p for some 0<p≤1.In particular, a growth functionΨ is equivalent to the function x↦∫0x(Ψ(s)/s)ds which is concave (see [17]). Now, for such a concave growth function of order p,(1.6)L1(Ω,P)⊂MΨ(Ω,P)=LΨ(Ω,P)=LΨ(Ω,P)⊂Lp(Ω,P),(1.7)‖f‖Ψ≲min⁡{∫ΩΨ(|f|)dP,(∫ΩΨ(|f|)dP)p},
while(1.8)‖f‖Ψ:=∫ΩΨ(|f|)dP≲max⁡{‖f‖Ψ,‖f‖Ψp}.
Moreover, if we define dΨ(f,g)=∥f-g∥Ψ (resp., dΨ(f,g)=∥f-g∥Ψ), then dΨ and dΨ (that we will simply denote by ∥·∥Ψ and ∥·∥Ψ) are two equivalent metrics on LΨ(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 theΔ2-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 thatΨ 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)=xp for 0<p≤1 and any x≥x0.
(2)Ψ2(x)=xplogq(C+x) (for x≥x0) with C>0 large enough, 0<p≤1 and q≥1.
(3)Ψ3(x)=Φp(x) at least for any x large, where Φ is an Orlicz function (see Remark 1.2) and p>0 is such that Φp(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.
LetΨ be as in (3) above. There exists a convex function ψ such that Ψ(x)=ψ(xp) for any x>0 large enough.Proof.
Letp>0 and Φ be an Orlicz function such that Ψ(x)=Φp(x) for any x≥0. Since Ψ is (assumed to be) bijective, we can define the function ψ by ψ(xp)=Φp(x) for every x. Now, using that Φ is convex, ψ is convex since
(1.9)ψ(x)x=(Φ(x1/p)x1/p)p≤(Φ(y1/p)y1/p)p=ψ(y)y,
for any 0≤x≤y.It follows thatΨ(|f|)=ψ(|f|p) is subharmonic (for f holomorphic).The following lemma gives an upper estimate of the Luxembourg norm of a function inL∞(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
(1.10)‖f‖Ψ≤‖f‖∞Ψ-1(‖f‖∞/‖f‖pp).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:
(1.11)∫ΩΨ(|f|C)dP≤∫Ω|f|pΨ(1C)dP≤‖f‖ppΨ(1C).
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

ForΨ a subharmonic-preserving growth function and α>-1, the weighted Bergman-Orlicz space AαΨ of the ball consists of those holomorphic functions on 𝔹N which belongs to the Orlicz space LΨ(𝔹N,vα). To avoid further confusion, we will denote by ∥·∥α,Ψ the corresponding Luxembourg (quasi)norm and by ∥·∥αΨ the quantity ∫𝔹NΨ(|f|)dvα. AαΨ is metric space for the distance dα,Ψ or dαΨ defined by, respectively, dα,Ψ(f,g)=∥f-g∥α,Ψ and dαΨ(f,g)=∥f-g∥αΨ. If Ψ(t)=tp, then we recover the usual weighted Bergman space Aαp. One checks that we have the followings inclusions:(1.12)Aα1⊂AαΨ⊂Aαp,
whenever Ψ is a growth function of order p.It seems important to us to mention that a linear operatorT from AαΨ1 to X with X=LΨ2(𝔹N,vα) or X=AαΨ2, 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 that(1.13)max⁡(‖T(f)‖α,Ψ2,‖T(f)‖αΨ2)≤C,
for any f∈AαΨ1 such that min(∥f∥α,Ψ1,∥f∥αΨ1)≤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 onAαΨ.Proposition 1.6.
Letα>-1 and let Ψ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈AαΨ, one has
(1.14)|f(a)|≤Ψ-1((21-|a|)(N+α+1))‖f‖α,Ψ.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 LΨ the following result.Corollary 1.7.
AαΨ, endowed with dα,Ψ or dαΨ, is a complete metric space.Letp>0 and α>-1. For a∈𝔹N, we introduce the following “test’’ function fa,α,p defined by(1.15)fa,α,p=(1-|a|2(1-〈z,a〉)2)(N+α+1)/p.|fa,α,p|p is nothing but the Berezin kernel, hence ∥fa,α,p∥∞=((1+|a|)/(1-|a|))(N+α+1)/p while ∥fa,α,p∥p=1. Then, as a consequence of Lemma 1.5, we have(1.16)‖fa,α,p‖α,Ψ≤(21-|a|)(N+α+1)/p1Ψ-1((2/1-|a|)(N+α+1)),
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 AαΨ. The definition needs to introduce the sets Q(ζ,h), defined by(1.17)Q(ζ,h)={z∈SN,|1-〈z,ζ〉|<h},
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 integerM>0 such that for any 0<r<1, one can find a finite sequence {ξk}k=1m (m depending on r) in 𝕊N with the following properties.(1)
𝕊N=⋃kQ(ξk,r).(2)
The setsQ(ξk,r/4) are mutually disjoint.(3)
Each point of𝕊N belongs to at most M of the sets Q(ξk,4r).From now on,M denotes the constant involved in the previous lemma. Let n≥0 be an integer and let Cn be the corona(1.18)Cn={z∈BN,1-12n≤|z|<1-12n+1}.
For any n≥0, let (ξn,k)k⊂𝕊N be given by Lemma 1.8 putting r=1/2n. For k≥0, we set(1.19)T0,k={z∈BN∖{0},z|z|∈Q(ξ0,k,1)}∪{0}.
Then we define the sets Tn,k, for n≥1 and k≥0, by(1.20)Tn,k={z∈BN∖{0},z|z|∈Q(ξn,k,12n)}.
We have both(1.21)⋃n≥0Cn=BN,⋃k≥0T0,k=BN,⋃k≥0Tn,k=BN∖{0},n≥1.
For (n,k)∈ℕ2, we finally define the subset Δ(n,k) of 𝔹N by Δ(n,k)=Cn∩Tn,k. These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function Λf for f∈AαΨ(𝔹N) by(1.22)Λf=∑n,k≥0sup⁡Δ(n,k)(|f(z)|)χΔ(n,k),
where χΔ(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.
LetΨ be a subharmonic-preserving growth function and let α>-1. Then the maximal operator Λ, which carries f to Λf, is bounded from AαΨ to LΨ(𝔹N,vα). More precisely, there exists B≥1 such that for every f∈AαΨ, one has
(1.23)‖Λf‖LΨ(BN,vα)≤B‖f‖α,Ψ.
In particular, a holomorphic function f belongs to AαΨ if and only if Λf belongs to LΨ(𝔹N,vα).


#### 1.2.2. Large Hardy-Orlicz Spaces

LetΨ still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let (Ω,ℙ)=(𝕊N,σ). The Hardy-Orlicz space HΨ of the ball consists of all holomorphic functions f on 𝔹N such that(1.24)‖f‖HΨ:=sup⁡0<r<1‖fr‖Ψ<∞,
where fr(z)=f(rz), and where ∥·∥Ψ is the Luxembourg norm on the Orlicz space LΨ(𝕊N,σ). Note that we can replace sup0<r<1∥fr∥Ψ<∞ by limr→1∥fr∥Ψ<∞ thanks to the subharmonicity of Ψ(|f|). Because Ψ is supposed to be a growth function, we have the following inclusion:(1.25)H1⊂HΨ⊂Hp
for a growth function Ψ of order p. In particular, every f∈HΨ admits a boundary radial limit, denoted by f*, σ-almost everywhere on 𝕊N. Let us also note that if Ψ is a growth function, then f∈HΨ if and only if(1.26)‖f‖HΨΨ:=sup⁡0<r<1‖fr‖Ψ=lim⁡r→1‖fr‖Ψ<∞(Inequalities(1.7)and(1.8)).Without possible confusion, we will write∥·∥Ψ instead of ∥·∥HΨ (resp., ∥·∥Ψ instead ∥·∥HΨΨ). As for Bergman-Orlicz spaces, a linear operator T from HΨ1 to some LΨ2(𝕊N,σ) or HΨ2 where Ψ1 and Ψ2 are two growth functions is continuous (bounded) if and only if there exists a constant C>0 such that(1.27)max⁡(‖T(f)‖Ψ2,‖T(f)‖Ψ2)≤C
for any f∈HΨ1 such that min(∥f∥Ψ1,∥f∥Ψ1)≤1.In addition, it is clear that, for anyα>-1,AαΨ⊃HΨ and ∥f∥α,Ψ≤∥f∥Ψ for any f∈HΨ. Therefore, letting α tend to −1 in Proposition 1.6, we get the following.Proposition 1.10.
LetΨ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈HΨ, one has
(1.28)|f(a)|≤Ψ-1((11-|a|)N)‖f‖Ψ.As a corollary, we have the following.Corollary 1.11.
LetΨ be a subharmonic-preserving growth function. HΨ is a complete metric space (with the equivalent distances induced by ∥·∥Ψ and ∥·∥Ψ, as usual).Fora∈𝔹N and p>0, we introduce the “test’’ function fa,p defined for any z∈𝔹N¯ by(1.29)fa,p(z)=(1-|a|1-〈z,a〉)2N/p.
It is easily seen that fa,p∈H∞ with ∥fa,p∥∞=1 and that |fa,p(z)|≤1 for any z∈𝕊N. Moreover, let us observe that(1.30)|fa,p(z)|p=(1-|a|1+|a|)NP(|a|z,a|a|),
so that ∥fa,p∥p=((1-|a|)/(1+|a|))N/p. Therefore, if Ψ is a growth function of order p, we have, by Lemma 1.5,(1.31)‖fa,p‖Ψ≤1Ψ-1(1/‖fa,p‖p)≤1Ψ-1((1/(1-|a|))N/p).It is very convenient to seeHΨ as a closed subspace of LΨ(𝕊N). 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 Hp with 0<p<1, even in this case. However, using a radial maximal function, we can still see Hp as a subspace of Lp. We are going to extend this to HΨ for Ψ a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on 𝔹N¯: for (z,w)∈𝔹N¯,(1.32)d(z,w)=|1-〈z,w〉|.
It is well known that d is a distance on 𝕊N and a pseudodistance on 𝔹N [21, Paragraph 5.1]. It permits to define the Korányi approach region Γ(ζ) for ζ∈𝕊N:(1.33)Γ(ζ)={z∈BN,d(z,ζ)2<1-|z|2}.
Then the maximal function Nf of f, associated to Korányi approach region, is given by(1.34)Nf(ζ)=sup⁡z∈Γ(ζ)|f(z)|
for any ζ∈𝕊N. [17, Theorem 1.3] will be very useful.Theorem 1.12.
LetΨ be a growth function. Then, for any f∈HΨ,
(1.35)∫SNΨ(|Nf|)dσ≲‖f‖Ψ.
In particular, a holomorphic function f belongs to HΨ if and only if Nf belongs to LΨ(𝕊N,σ).From this theorem, we deduce the following one.Theorem 1.13.
LetΨ be a subharmonic-preserving growth function. Then for every f∈HΨ, one has(1)
limr→1∫𝕊NΨ(|f*-fr|)dσ=0;(2)
∥f*∥Ψ=∥f∥HΨΨ;(3)
HΨ is separable. More precisely, the polynomials are dense in HΨ.Proof.
LetMradf(ζ)=sup0<r<1|f(rζ)| for ζ∈𝕊N. Obviously, Mradf≤Nf, hence ∫𝕊NΨ(Mradf)dσ≲∥f∥Ψ (Theorem 1.12). Since Ψ is concave and vanishes at 0, we have Ψ(|f*-fr|)≤Ψ(|f*|)+Ψ(|fr|)≤2Ψ(Mradf). Now Ψ(|f*-fr|(ζ)) tends to 0 as r goes to 1 for σ-almost every ζ∈𝕊N. By the dominated convergence theorem, (1) follows.
Then∥f*∥Ψ=limr→1∥fr∥Ψ and (2) comes from to the subharmonicity of Ψ(|f|).
We proved in(1) that fr tends to f* in HΨ for ∥·∥Ψ (hence for ∥·∥Ψ also). We approach every fr uniformly on 𝔹N¯ by its Taylor series to get the third assertion.


## 1.1. Introduction

Let𝔹N={z=(z1,…,zN)∈ℂN:|z|2=∑i=1N|zi|2<1} and 𝕊N=∂𝔹N, N≥1, denoting, respectively, the unit ball and the unit sphere of ℂN. For N=1, we denote by 𝔻 the unit disc of the complex plane.For a large class of spacesX of holomorphic functions in the unit disc or the unit ball, characterizations of the boundedness and compactness of the canonical embedding X↪Lp(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, whenX=Hp(𝔻), Carleson [1] proved that Hp(𝔻)↪Lp(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 Hp(𝔻)↪Lq(μ), with 0<p≤q<∞, in terms of (q/p)-Carleson measures. For the unweighted and weighted Bergman spaces Aαp(𝔹N), N≥1, similar results were obtained by Cima and Wogen [5], Luecking [6], Ueki [7]. We recall also that the compactness of X↪Lp(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 𝔹N¯, then μ∣𝕊N is absolutely continuous with respect to 𝕊N. 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 ofHp(𝔹N)↪Lq(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 𝕊N 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 Hp and H∞.This motivation is reinforced by a second observation: if the measureμϕ is the pull-back measure of the invariant-rotation measure on 𝕊N under a holomorphic map ϕ:𝔹N→𝔹N, N≥1, then μϕ 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 Hp and H∞ for any N≥1. This observation is directly connected to the study of composition operators Cϕ on Hp, which are defined by Cϕ(f)=f∘ϕ for f∈H(𝔹N), and which may be seen as the embedding operators Hp↪Lp(μϕ). By the way, this leads the authors of [9, 10] to state Carleson theorems for Hardy-Orlicz and Bergman-Orlicz spaces (resp., denoted by HΨ(𝔻) and AΨ(𝔻)) in the unit disc, when the defining function Ψ is an Orlicz function. These spaces appear as good candidates for generalizing Hp and Ap 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=μϕ is the pull-back measure under a holomorphic self-map ϕ of 𝔹N, then HΨ always embeds into LΨ(μ) whenever Ψ satisfied a fast growth condition (namely, the Δ2-Condition, that implies Ψ(x)≥ex for large value of x and means that we are close to H∞). Similarly, it was shown in [13] that if HΨ compactly embeds into LΨ(μ) for everyΨ, then H∞↪L∞(μ) compactly (the converse easily holds). We mention that there is no Ψ such that HΨ↪LΨ(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 HΨ↪LΨ(μ) boundedly (resp., compactly) if and only if Hp↪Lp(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 thanH1(𝔹N). 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 HΨ(𝔹N)↪Lq(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 AαΨ(𝔻), where Ψ=log+p, 1≤p<∞, that is, when AαΨ(𝔻) is the area Nevanlinna space. These measures reveal to be those μ for which Aαp↪Lp(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 thegood 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 HΨ1(𝔹N)↪LΨ2(𝔹N,μ) (resp., AαΨ1(𝔹N)↪LΨ2(𝔹N,μ)) is bounded or compact, when Ψ1 and Ψ2 are two growth functions (i.e., for which H1⊂HΨi⊂Hp (resp., Aα1⊂AαΨi⊂Aαp) 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 Aα1, 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 Aα1 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∈ℂN, the euclidean inner product of z and w will be denoted by 〈z,w〉, that is, 〈z,w〉=∑i=1Nziwi¯; 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,vα will be the measure on 𝔹N defined by dvα=cα(1-|z|)αdv, where v is the Lebesgue measure on 𝔹N and cα 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 writeHψ (resp., Aαψ) instead of Hψ(𝔹N) (resp., Aαψ(𝔹N)).


## 1.2. Preliminaries: Menagerie of Spaces

LetΨ:[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 LΨ(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 MΨ(Ω,ℙ) by(1.1)MΨ(Ω,P)={f:Ω⟶Cmeasurable;∫ΩΨ(|f|C)<∞foranyC>0},
and we also introduce the following set:(1.2)LΨ(Ω,P)={f:Ω⟶Cmeasurable;∫ΩΨ(|f|)<∞}.
In general, these three sets are not vector spaces and do not coincide, but we trivially have(1.3)MΨ(Ω,P)⊂LΨ(Ω,P)⊂LΨ(Ω,P).
We also define the Luxembourg gauge on LΨ(Ω,ℙ) by(1.4)‖f‖Ψ=inf⁡(λ>0,∫ΩΨ(|f|λ)≤1).
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 functionsΨ1 and Ψ2 as above are equivalent if there exists some constant c such that(1.5)cΨ1(cx)≤Ψ2(x)≤c-1Ψ1(c-1x),
for any x large enough. Two equivalent functions define the same Orlicz class with equivalent Luxembourg functionals.In order to define a good topology onLΨ(Ω,ℙ) and to get properties convenient for our purpose, we will assume that Ψ satisfies the following definition.Definition 1.1.
Let0<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)≤ypΨ(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 thatψ is a growth function if it is a growth function of order p for some 0<p≤1.In particular, a growth functionΨ is equivalent to the function x↦∫0x(Ψ(s)/s)ds which is concave (see [17]). Now, for such a concave growth function of order p,(1.6)L1(Ω,P)⊂MΨ(Ω,P)=LΨ(Ω,P)=LΨ(Ω,P)⊂Lp(Ω,P),(1.7)‖f‖Ψ≲min⁡{∫ΩΨ(|f|)dP,(∫ΩΨ(|f|)dP)p},
while(1.8)‖f‖Ψ:=∫ΩΨ(|f|)dP≲max⁡{‖f‖Ψ,‖f‖Ψp}.
Moreover, if we define dΨ(f,g)=∥f-g∥Ψ (resp., dΨ(f,g)=∥f-g∥Ψ), then dΨ and dΨ (that we will simply denote by ∥·∥Ψ and ∥·∥Ψ) are two equivalent metrics on LΨ(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 theΔ2-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 thatΨ 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)=xp for 0<p≤1 and any x≥x0.
(2)Ψ2(x)=xplogq(C+x) (for x≥x0) with C>0 large enough, 0<p≤1 and q≥1.
(3)Ψ3(x)=Φp(x) at least for any x large, where Φ is an Orlicz function (see Remark 1.2) and p>0 is such that Φp(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.
LetΨ be as in (3) above. There exists a convex function ψ such that Ψ(x)=ψ(xp) for any x>0 large enough.Proof.
Letp>0 and Φ be an Orlicz function such that Ψ(x)=Φp(x) for any x≥0. Since Ψ is (assumed to be) bijective, we can define the function ψ by ψ(xp)=Φp(x) for every x. Now, using that Φ is convex, ψ is convex since
(1.9)ψ(x)x=(Φ(x1/p)x1/p)p≤(Φ(y1/p)y1/p)p=ψ(y)y,
for any 0≤x≤y.It follows thatΨ(|f|)=ψ(|f|p) is subharmonic (for f holomorphic).The following lemma gives an upper estimate of the Luxembourg norm of a function inL∞(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
(1.10)‖f‖Ψ≤‖f‖∞Ψ-1(‖f‖∞/‖f‖pp).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:
(1.11)∫ΩΨ(|f|C)dP≤∫Ω|f|pΨ(1C)dP≤‖f‖ppΨ(1C).
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

ForΨ a subharmonic-preserving growth function and α>-1, the weighted Bergman-Orlicz space AαΨ of the ball consists of those holomorphic functions on 𝔹N which belongs to the Orlicz space LΨ(𝔹N,vα). To avoid further confusion, we will denote by ∥·∥α,Ψ the corresponding Luxembourg (quasi)norm and by ∥·∥αΨ the quantity ∫𝔹NΨ(|f|)dvα. AαΨ is metric space for the distance dα,Ψ or dαΨ defined by, respectively, dα,Ψ(f,g)=∥f-g∥α,Ψ and dαΨ(f,g)=∥f-g∥αΨ. If Ψ(t)=tp, then we recover the usual weighted Bergman space Aαp. One checks that we have the followings inclusions:(1.12)Aα1⊂AαΨ⊂Aαp,
whenever Ψ is a growth function of order p.It seems important to us to mention that a linear operatorT from AαΨ1 to X with X=LΨ2(𝔹N,vα) or X=AαΨ2, 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 that(1.13)max⁡(‖T(f)‖α,Ψ2,‖T(f)‖αΨ2)≤C,
for any f∈AαΨ1 such that min(∥f∥α,Ψ1,∥f∥αΨ1)≤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 onAαΨ.Proposition 1.6.
Letα>-1 and let Ψ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈AαΨ, one has
(1.14)|f(a)|≤Ψ-1((21-|a|)(N+α+1))‖f‖α,Ψ.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 LΨ the following result.Corollary 1.7.
AαΨ, endowed with dα,Ψ or dαΨ, is a complete metric space.Letp>0 and α>-1. For a∈𝔹N, we introduce the following “test’’ function fa,α,p defined by(1.15)fa,α,p=(1-|a|2(1-〈z,a〉)2)(N+α+1)/p.|fa,α,p|p is nothing but the Berezin kernel, hence ∥fa,α,p∥∞=((1+|a|)/(1-|a|))(N+α+1)/p while ∥fa,α,p∥p=1. Then, as a consequence of Lemma 1.5, we have(1.16)‖fa,α,p‖α,Ψ≤(21-|a|)(N+α+1)/p1Ψ-1((2/1-|a|)(N+α+1)),
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 AαΨ. The definition needs to introduce the sets Q(ζ,h), defined by(1.17)Q(ζ,h)={z∈SN,|1-〈z,ζ〉|<h},
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 integerM>0 such that for any 0<r<1, one can find a finite sequence {ξk}k=1m (m depending on r) in 𝕊N with the following properties.(1)
𝕊N=⋃kQ(ξk,r).(2)
The setsQ(ξk,r/4) are mutually disjoint.(3)
Each point of𝕊N belongs to at most M of the sets Q(ξk,4r).From now on,M denotes the constant involved in the previous lemma. Let n≥0 be an integer and let Cn be the corona(1.18)Cn={z∈BN,1-12n≤|z|<1-12n+1}.
For any n≥0, let (ξn,k)k⊂𝕊N be given by Lemma 1.8 putting r=1/2n. For k≥0, we set(1.19)T0,k={z∈BN∖{0},z|z|∈Q(ξ0,k,1)}∪{0}.
Then we define the sets Tn,k, for n≥1 and k≥0, by(1.20)Tn,k={z∈BN∖{0},z|z|∈Q(ξn,k,12n)}.
We have both(1.21)⋃n≥0Cn=BN,⋃k≥0T0,k=BN,⋃k≥0Tn,k=BN∖{0},n≥1.
For (n,k)∈ℕ2, we finally define the subset Δ(n,k) of 𝔹N by Δ(n,k)=Cn∩Tn,k. These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function Λf for f∈AαΨ(𝔹N) by(1.22)Λf=∑n,k≥0sup⁡Δ(n,k)(|f(z)|)χΔ(n,k),
where χΔ(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.
LetΨ be a subharmonic-preserving growth function and let α>-1. Then the maximal operator Λ, which carries f to Λf, is bounded from AαΨ to LΨ(𝔹N,vα). More precisely, there exists B≥1 such that for every f∈AαΨ, one has
(1.23)‖Λf‖LΨ(BN,vα)≤B‖f‖α,Ψ.
In particular, a holomorphic function f belongs to AαΨ if and only if Λf belongs to LΨ(𝔹N,vα).


### 1.2.2. Large Hardy-Orlicz Spaces

LetΨ still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let (Ω,ℙ)=(𝕊N,σ). The Hardy-Orlicz space HΨ of the ball consists of all holomorphic functions f on 𝔹N such that(1.24)‖f‖HΨ:=sup⁡0<r<1‖fr‖Ψ<∞,
where fr(z)=f(rz), and where ∥·∥Ψ is the Luxembourg norm on the Orlicz space LΨ(𝕊N,σ). Note that we can replace sup0<r<1∥fr∥Ψ<∞ by limr→1∥fr∥Ψ<∞ thanks to the subharmonicity of Ψ(|f|). Because Ψ is supposed to be a growth function, we have the following inclusion:(1.25)H1⊂HΨ⊂Hp
for a growth function Ψ of order p. In particular, every f∈HΨ admits a boundary radial limit, denoted by f*, σ-almost everywhere on 𝕊N. Let us also note that if Ψ is a growth function, then f∈HΨ if and only if(1.26)‖f‖HΨΨ:=sup⁡0<r<1‖fr‖Ψ=lim⁡r→1‖fr‖Ψ<∞(Inequalities(1.7)and(1.8)).Without possible confusion, we will write∥·∥Ψ instead of ∥·∥HΨ (resp., ∥·∥Ψ instead ∥·∥HΨΨ). As for Bergman-Orlicz spaces, a linear operator T from HΨ1 to some LΨ2(𝕊N,σ) or HΨ2 where Ψ1 and Ψ2 are two growth functions is continuous (bounded) if and only if there exists a constant C>0 such that(1.27)max⁡(‖T(f)‖Ψ2,‖T(f)‖Ψ2)≤C
for any f∈HΨ1 such that min(∥f∥Ψ1,∥f∥Ψ1)≤1.In addition, it is clear that, for anyα>-1,AαΨ⊃HΨ and ∥f∥α,Ψ≤∥f∥Ψ for any f∈HΨ. Therefore, letting α tend to −1 in Proposition 1.6, we get the following.Proposition 1.10.
LetΨ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈HΨ, one has
(1.28)|f(a)|≤Ψ-1((11-|a|)N)‖f‖Ψ.As a corollary, we have the following.Corollary 1.11.
LetΨ be a subharmonic-preserving growth function. HΨ is a complete metric space (with the equivalent distances induced by ∥·∥Ψ and ∥·∥Ψ, as usual).Fora∈𝔹N and p>0, we introduce the “test’’ function fa,p defined for any z∈𝔹N¯ by(1.29)fa,p(z)=(1-|a|1-〈z,a〉)2N/p.
It is easily seen that fa,p∈H∞ with ∥fa,p∥∞=1 and that |fa,p(z)|≤1 for any z∈𝕊N. Moreover, let us observe that(1.30)|fa,p(z)|p=(1-|a|1+|a|)NP(|a|z,a|a|),
so that ∥fa,p∥p=((1-|a|)/(1+|a|))N/p. Therefore, if Ψ is a growth function of order p, we have, by Lemma 1.5,(1.31)‖fa,p‖Ψ≤1Ψ-1(1/‖fa,p‖p)≤1Ψ-1((1/(1-|a|))N/p).It is very convenient to seeHΨ as a closed subspace of LΨ(𝕊N). 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 Hp with 0<p<1, even in this case. However, using a radial maximal function, we can still see Hp as a subspace of Lp. We are going to extend this to HΨ for Ψ a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on 𝔹N¯: for (z,w)∈𝔹N¯,(1.32)d(z,w)=|1-〈z,w〉|.
It is well known that d is a distance on 𝕊N and a pseudodistance on 𝔹N [21, Paragraph 5.1]. It permits to define the Korányi approach region Γ(ζ) for ζ∈𝕊N:(1.33)Γ(ζ)={z∈BN,d(z,ζ)2<1-|z|2}.
Then the maximal function Nf of f, associated to Korányi approach region, is given by(1.34)Nf(ζ)=sup⁡z∈Γ(ζ)|f(z)|
for any ζ∈𝕊N. [17, Theorem 1.3] will be very useful.Theorem 1.12.
LetΨ be a growth function. Then, for any f∈HΨ,
(1.35)∫SNΨ(|Nf|)dσ≲‖f‖Ψ.
In particular, a holomorphic function f belongs to HΨ if and only if Nf belongs to LΨ(𝕊N,σ).From this theorem, we deduce the following one.Theorem 1.13.
LetΨ be a subharmonic-preserving growth function. Then for every f∈HΨ, one has(1)
limr→1∫𝕊NΨ(|f*-fr|)dσ=0;(2)
∥f*∥Ψ=∥f∥HΨΨ;(3)
HΨ is separable. More precisely, the polynomials are dense in HΨ.Proof.
LetMradf(ζ)=sup0<r<1|f(rζ)| for ζ∈𝕊N. Obviously, Mradf≤Nf, hence ∫𝕊NΨ(Mradf)dσ≲∥f∥Ψ (Theorem 1.12). Since Ψ is concave and vanishes at 0, we have Ψ(|f*-fr|)≤Ψ(|f*|)+Ψ(|fr|)≤2Ψ(Mradf). Now Ψ(|f*-fr|(ζ)) tends to 0 as r goes to 1 for σ-almost every ζ∈𝕊N. By the dominated convergence theorem, (1) follows.
Then∥f*∥Ψ=limr→1∥fr∥Ψ and (2) comes from to the subharmonicity of Ψ(|f|).
We proved in(1) that fr tends to f* in HΨ for ∥·∥Ψ (hence for ∥·∥Ψ also). We approach every fr uniformly on 𝔹N¯ by its Taylor series to get the third assertion.


## 1.2.1. Large Bergman-Orlicz Spaces

ForΨ a subharmonic-preserving growth function and α>-1, the weighted Bergman-Orlicz space AαΨ of the ball consists of those holomorphic functions on 𝔹N which belongs to the Orlicz space LΨ(𝔹N,vα). To avoid further confusion, we will denote by ∥·∥α,Ψ the corresponding Luxembourg (quasi)norm and by ∥·∥αΨ the quantity ∫𝔹NΨ(|f|)dvα. AαΨ is metric space for the distance dα,Ψ or dαΨ defined by, respectively, dα,Ψ(f,g)=∥f-g∥α,Ψ and dαΨ(f,g)=∥f-g∥αΨ. If Ψ(t)=tp, then we recover the usual weighted Bergman space Aαp. One checks that we have the followings inclusions:(1.12)Aα1⊂AαΨ⊂Aαp,
whenever Ψ is a growth function of order p.It seems important to us to mention that a linear operatorT from AαΨ1 to X with X=LΨ2(𝔹N,vα) or X=AαΨ2, 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 that(1.13)max⁡(‖T(f)‖α,Ψ2,‖T(f)‖αΨ2)≤C,
for any f∈AαΨ1 such that min(∥f∥α,Ψ1,∥f∥αΨ1)≤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 onAαΨ.Proposition 1.6.
Letα>-1 and let Ψ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈AαΨ, one has
(1.14)|f(a)|≤Ψ-1((21-|a|)(N+α+1))‖f‖α,Ψ.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 LΨ the following result.Corollary 1.7.
AαΨ, endowed with dα,Ψ or dαΨ, is a complete metric space.Letp>0 and α>-1. For a∈𝔹N, we introduce the following “test’’ function fa,α,p defined by(1.15)fa,α,p=(1-|a|2(1-〈z,a〉)2)(N+α+1)/p.|fa,α,p|p is nothing but the Berezin kernel, hence ∥fa,α,p∥∞=((1+|a|)/(1-|a|))(N+α+1)/p while ∥fa,α,p∥p=1. Then, as a consequence of Lemma 1.5, we have(1.16)‖fa,α,p‖α,Ψ≤(21-|a|)(N+α+1)/p1Ψ-1((2/1-|a|)(N+α+1)),
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 AαΨ. The definition needs to introduce the sets Q(ζ,h), defined by(1.17)Q(ζ,h)={z∈SN,|1-〈z,ζ〉|<h},
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 integerM>0 such that for any 0<r<1, one can find a finite sequence {ξk}k=1m (m depending on r) in 𝕊N with the following properties.(1)
𝕊N=⋃kQ(ξk,r).(2)
The setsQ(ξk,r/4) are mutually disjoint.(3)
Each point of𝕊N belongs to at most M of the sets Q(ξk,4r).From now on,M denotes the constant involved in the previous lemma. Let n≥0 be an integer and let Cn be the corona(1.18)Cn={z∈BN,1-12n≤|z|<1-12n+1}.
For any n≥0, let (ξn,k)k⊂𝕊N be given by Lemma 1.8 putting r=1/2n. For k≥0, we set(1.19)T0,k={z∈BN∖{0},z|z|∈Q(ξ0,k,1)}∪{0}.
Then we define the sets Tn,k, for n≥1 and k≥0, by(1.20)Tn,k={z∈BN∖{0},z|z|∈Q(ξn,k,12n)}.
We have both(1.21)⋃n≥0Cn=BN,⋃k≥0T0,k=BN,⋃k≥0Tn,k=BN∖{0},n≥1.
For (n,k)∈ℕ2, we finally define the subset Δ(n,k) of 𝔹N by Δ(n,k)=Cn∩Tn,k. These sets have good covering properties that we do not recall here (we refer to [11]). Anyway, we define the following maximal function Λf for f∈AαΨ(𝔹N) by(1.22)Λf=∑n,k≥0sup⁡Δ(n,k)(|f(z)|)χΔ(n,k),
where χΔ(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.
LetΨ be a subharmonic-preserving growth function and let α>-1. Then the maximal operator Λ, which carries f to Λf, is bounded from AαΨ to LΨ(𝔹N,vα). More precisely, there exists B≥1 such that for every f∈AαΨ, one has
(1.23)‖Λf‖LΨ(BN,vα)≤B‖f‖α,Ψ.
In particular, a holomorphic function f belongs to AαΨ if and only if Λf belongs to LΨ(𝔹N,vα).


## 1.2.2. Large Hardy-Orlicz Spaces

LetΨ still be a (concave) subharmonic-preserving growth function. With the notations of Section 1.2, let (Ω,ℙ)=(𝕊N,σ). The Hardy-Orlicz space HΨ of the ball consists of all holomorphic functions f on 𝔹N such that(1.24)‖f‖HΨ:=sup⁡0<r<1‖fr‖Ψ<∞,
where fr(z)=f(rz), and where ∥·∥Ψ is the Luxembourg norm on the Orlicz space LΨ(𝕊N,σ). Note that we can replace sup0<r<1∥fr∥Ψ<∞ by limr→1∥fr∥Ψ<∞ thanks to the subharmonicity of Ψ(|f|). Because Ψ is supposed to be a growth function, we have the following inclusion:(1.25)H1⊂HΨ⊂Hp
for a growth function Ψ of order p. In particular, every f∈HΨ admits a boundary radial limit, denoted by f*, σ-almost everywhere on 𝕊N. Let us also note that if Ψ is a growth function, then f∈HΨ if and only if(1.26)‖f‖HΨΨ:=sup⁡0<r<1‖fr‖Ψ=lim⁡r→1‖fr‖Ψ<∞(Inequalities(1.7)and(1.8)).Without possible confusion, we will write∥·∥Ψ instead of ∥·∥HΨ (resp., ∥·∥Ψ instead ∥·∥HΨΨ). As for Bergman-Orlicz spaces, a linear operator T from HΨ1 to some LΨ2(𝕊N,σ) or HΨ2 where Ψ1 and Ψ2 are two growth functions is continuous (bounded) if and only if there exists a constant C>0 such that(1.27)max⁡(‖T(f)‖Ψ2,‖T(f)‖Ψ2)≤C
for any f∈HΨ1 such that min(∥f∥Ψ1,∥f∥Ψ1)≤1.In addition, it is clear that, for anyα>-1,AαΨ⊃HΨ and ∥f∥α,Ψ≤∥f∥Ψ for any f∈HΨ. Therefore, letting α tend to −1 in Proposition 1.6, we get the following.Proposition 1.10.
LetΨ be a subharmonic-preserving growth function. For any a∈𝔹N and any f∈HΨ, one has
(1.28)|f(a)|≤Ψ-1((11-|a|)N)‖f‖Ψ.As a corollary, we have the following.Corollary 1.11.
LetΨ be a subharmonic-preserving growth function. HΨ is a complete metric space (with the equivalent distances induced by ∥·∥Ψ and ∥·∥Ψ, as usual).Fora∈𝔹N and p>0, we introduce the “test’’ function fa,p defined for any z∈𝔹N¯ by(1.29)fa,p(z)=(1-|a|1-〈z,a〉)2N/p.
It is easily seen that fa,p∈H∞ with ∥fa,p∥∞=1 and that |fa,p(z)|≤1 for any z∈𝕊N. Moreover, let us observe that(1.30)|fa,p(z)|p=(1-|a|1+|a|)NP(|a|z,a|a|),
so that ∥fa,p∥p=((1-|a|)/(1+|a|))N/p. Therefore, if Ψ is a growth function of order p, we have, by Lemma 1.5,(1.31)‖fa,p‖Ψ≤1Ψ-1(1/‖fa,p‖p)≤1Ψ-1((1/(1-|a|))N/p).It is very convenient to seeHΨ as a closed subspace of LΨ(𝕊N). 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 Hp with 0<p<1, even in this case. However, using a radial maximal function, we can still see Hp as a subspace of Lp. We are going to extend this to HΨ for Ψ a growth function which preserves the subharmonicity. To this purpose, we recall the definition of the nonisotropic distance on 𝔹N¯: for (z,w)∈𝔹N¯,(1.32)d(z,w)=|1-〈z,w〉|.
It is well known that d is a distance on 𝕊N and a pseudodistance on 𝔹N [21, Paragraph 5.1]. It permits to define the Korányi approach region Γ(ζ) for ζ∈𝕊N:(1.33)Γ(ζ)={z∈BN,d(z,ζ)2<1-|z|2}.
Then the maximal function Nf of f, associated to Korányi approach region, is given by(1.34)Nf(ζ)=sup⁡z∈Γ(ζ)|f(z)|
for any ζ∈𝕊N. [17, Theorem 1.3] will be very useful.Theorem 1.12.
LetΨ be a growth function. Then, for any f∈HΨ,
(1.35)∫SNΨ(|Nf|)dσ≲‖f‖Ψ.
In particular, a holomorphic function f belongs to HΨ if and only if Nf belongs to LΨ(𝕊N,σ).From this theorem, we deduce the following one.Theorem 1.13.
LetΨ be a subharmonic-preserving growth function. Then for every f∈HΨ, one has(1)
limr→1∫𝕊NΨ(|f*-fr|)dσ=0;(2)
∥f*∥Ψ=∥f∥HΨΨ;(3)
HΨ is separable. More precisely, the polynomials are dense in HΨ.Proof.
LetMradf(ζ)=sup0<r<1|f(rζ)| for ζ∈𝕊N. Obviously, Mradf≤Nf, hence ∫𝕊NΨ(Mradf)dσ≲∥f∥Ψ (Theorem 1.12). Since Ψ is concave and vanishes at 0, we have Ψ(|f*-fr|)≤Ψ(|f*|)+Ψ(|fr|)≤2Ψ(Mradf). Now Ψ(|f*-fr|(ζ)) tends to 0 as r goes to 1 for σ-almost every ζ∈𝕊N. By the dominated convergence theorem, (1) follows.
Then∥f*∥Ψ=limr→1∥fr∥Ψ and (2) comes from to the subharmonicity of Ψ(|f|).
We proved in(1) that fr tends to f* in HΨ for ∥·∥Ψ (hence for ∥·∥Ψ also). We approach every fr uniformly on 𝔹N¯ by its Taylor series to get the third assertion.


## 2. Carleson Embedding Theorems

### 2.1. Statements of the Results

Forζ∈𝔹N¯ and h∈]0,1], we define the nonisotropic “ball’’ of 𝔹N by(2.1)S(ζ,h)={z∈BN,d(ζ,z)2<h}
and its analogue in 𝔹N¯ by(2.2)S(ζ,h)={z∈BN¯,d(ζ,z)2<h}.
Let us also denote by(2.3)Q(ζ,h)=S(ζ,h)∩SN,
the “true’’ balls in 𝕊N. We have σ(Q(ζ,h))≈hN and vα(S(ζ,h))≈hN+α+1 [22].Letμ be a positive Borel measure on 𝔹N¯ whose restriction to 𝕊N 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))≤ChNq/p, while it is a vanishing (q/p)-Carleson measure if μ(𝒮(ζ,h))=o(hNq/p) when h goes to 0. A variant of the well-known Carleson theorem for Hardy spaces [1, 3] ensures that the embedding Hp↪Lq(𝔹N¯,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 Aαp↪Lq(𝔹N,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 thanH1 (resp., Aα1) (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 thanH1 (resp., Aα1), we state that the characterizations of the boundedness and compactness of HΨ1↪LΨ2(𝔹N¯,μ) (resp., AαΨ1↪LΨ2(𝔹N,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 ifΨi(x)=xpi, Ψi=Φpi (with Φ an Orlicz function), or Ψi(x)=xpilogqi(C+x) with p2≥p1, then x↦Ψ2∘Ψ1-1(x)/x is nondecreasing.Theorem 2.1.
LetΨ1 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 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ). Then,(1)
HΨ1 embeds into LΨ2(𝔹N¯,μ) if and only if there exists some h0∈(0,1) such that, for any h∈(0,h0),
(2.4)μ(S(ζ,h))≲1Ψ2∘Ψ1-1(1/hN)
uniformly in ζ∈𝕊N,(2)
the embeddingHΨ1↪LΨ2(𝔹N¯,μ) is compact if and only if
(2.5)μ(S(ζ,h))=oh→0(1Ψ2∘Ψ1-1(1/hN))
uniformly in ζ∈𝕊N.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.
LetΨ1 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 𝔹N. Then,(1)
AαΨ1 embeds into LΨ2(𝔹N,μ) if and only if there exists h0∈(0,1) such that, for any h∈(0,h0),
(2.6)μ(S(ζ,h))≲1Ψ2∘Ψ1-1(1/hN+α+1),
uniformly in ζ∈𝕊N,(2)
the embeddingAαΨ1↪LΨ2(𝔹N,μ) is compact if and only if
(2.7)μ(S(ζ,h))≤oh→0(1Ψ2∘Ψ1-1(1/hN+α+1)),
uniformly in ζ∈𝕊N.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.
Letα>-1 and let 0<p≤q<1 and Ψ1=Φp, Ψ2=Φq as in (3) of Example 1.3. Let μ be a finite positive Borel measure on 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ) (resp., on 𝔹N). Then,(1)
HΦp (resp., AαΦp) embeds into LΦq(𝔹N¯,μ) (resp., LΦq(𝔹N,μ)) if and only if μ is a (q/p)-Carleson measure (resp., a (α,(q/p))-Bergman-Carleson measure),(2)
the embeddingHΦp↪LΦq(𝔹N¯,μ) (resp., AαΦp↪LΦq(𝔹N,μ)) is compact if and only if μ is a vanishing (q/p)-Carleson measure (resp., a vanishing (α,(q/p)) Bergman-Carleson measure).IfΨ1=Ψ2 (or equivalently if Ψ1 and Ψ2 are equivalent), we have the following.Corollary 2.5.
Letα>-1, and let Ψ be subharmonic-preserving growth function. Let μ be a finite positive Borel measure on 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ) (resp., on 𝔹N). Then,(1)
HΨ (resp., AαΨ) embeds into LΨ(𝔹N¯,μ) (resp., LΨ(𝔹N,μ)) if and only if μ is a Carleson measure (resp., a α-Bergman-Carleson measure),(2)
the embeddingHΨ↪LΨ(𝔹N¯,μ) (resp., AαΨ↪LΨ(𝔹N,μ)) is compact if and only if μ is a vanishing Carleson measure (resp., a vanishing α-Bergman-Carleson measure).


### 2.2. Proofs of Theorems2.1 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 HΨ (resp., AαΨ), for Ψ  a subharmonic-preserving growth function, implies the convergence on every compact subset of 𝔹N, but it stems from Proposition 1.10 (resp., Proposition 1.6).Proposition 2.6.
Letα>-1, let Ψ1 and Ψ2 be two subharmonic-preserving growth functions and let μ (resp., μ) be a finite positive Borel measure on 𝔹N¯ (resp., 𝔹N) whose restriction to 𝕊N is absolutely continuous with respect to σ. One assume that jμ:HΨ1↪LΨ2(𝔹N¯,μ) (resp., jμα:AαΨ1↪LΨ2(𝔹N,μ)) is well defined (hence bounded).(1)
The two following assertions are equivalent:
(a)
the canonical embeddingjμ (resp., jμα) is compact;(b)
every sequence in the unit ball ofHΨ1 (resp., AαΨ1), which is convergent to 0 uniformly on every compact subset of 𝔹N, is convergent to 0 in LΨ2(μ) (resp., LΨ2(μ)).(2)
Iflimr→1-∥Ir∥=0 (resp., limr→1-∥Ir,α∥=0), where Ir(f)=f·χ𝔹N¯∖r𝔹N (resp., Ir(f)=f·χ𝔹N∖r𝔹N), then the canonical embedding jμ (resp., jμα) is compact.


#### 2.2.1. Proof of Theorem2.1

We assume that the hypothesis of Theorem2.1 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 Kμ, associated to μ by(2.8)Kμ(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN,forh∈(0,1),
so that μ is a Carleson measure if and only if Kμ(h) is finite for some h∈(0,1).Theorem 2.7 (see [12, Theorem 2.4]).
There exist two constantsC̃>0 and C>1 such that, for every f continuous on 𝔹N which admits boundary values almost everywhere on 𝕊N, and every finite positive Borel measure μ on 𝔹N¯, one has
(2.9)μ({z∈BN¯,|z|>1-h,|f(z)|>t})≤C̃Kμ(Ch)σ({Nf>t}),
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.
Letμ be a finite positive Borel measure on 𝔹N¯ 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 hA∈(0,1/C) such that
(2.10)Kμ(h)≤η1/hNΨ2(AΨ1-1(1/hN)),
for every h∈(0,hA). Then, for every f∈HΨ1(𝔹N) such that ∥f∥Ψ1≤1 and every Borel subset E of 𝔹N¯,
(2.11)∫EΨ2(|f|)dμ≲μ(E)+η∫SNΨ1(Nf)dσ,
where ≲ involves a constant which is independent of f, η, and E.Proof.
Letf∈HΨ1(𝔹N) 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. 											(2.12)∫EΨ2(A2N+1(C+1)CN-1|f|)dμ≤∫0xAΨ2′(s)μ(E)ds+ηC̃CN∫xA∞Ψ2′(s)Ψ1((((C+1)CN-1)/A)S)Ψ2(((C+1)/C)S)×σ({Nf>2N+1(C+1)CN-1As})ds,
for some constant xA which depends only on A. Now, since Ψ1 and Ψ2 are two concave growth functions, we have
(2.13)Ψi(s)≲Ψi(Ks)≲Ψi(s),
where ≲ involve constants which only depend on K and Ψi. Therefore, (2.12) becomes
(2.14)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ2′(s)Ψ1(s)Ψ2(s)σ({Nf>2N+1(C+1)CN-1As})ds.
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,
(2.15)Ψ1′(s)Ψ2′(s)≤Ψ1(s)Ψ2(s).
It follows that
(2.16)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ1′(s)σ({Nf>2N+1(C+1)CN-1As})ds≲μ(E)+η∫0+∞Ψ1({ANf2N+1(C+1)CN-1>s})ds=μ(E)+η∫SNψ1(ANf2N+1(C+1)CN-1)dσ≲μ(E)+η∫SNψ1(Nf)dσ.We now can prove Theorem2.1.Proof of Theorem2.1.
(1) We assume that μ is a (Ψ1,Ψ2)-Carleson measure, and we intend to show that jμ:HΨ1↪LΨ2(𝔹N¯,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),
(2.17)μ(S(ζ,h))hN≲1/hNΨ2∘Ψ1-1(1/hN),
uniformly in ζ∈𝕊N. 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),
(2.18)Kμ(h)≲1/hNΨ2∘Ψ1-1(1/hN).
Therefore, we may and will apply Lemma 2.8 with E=𝔹N¯ to get
(2.19)∫BN¯Ψ2(|f|)dμ≲μ(E)+∫SNΨ1(Nf)dσ.
We finish the proof of (1) using Theorem 1.12 and Inequality (1.7):
(2.20)∫BN¯Ψ2(|f|)dμ≲μ(E)+‖f‖Ψ1.
For the converse, letga,p=Φ-1((1/(1-|a|))N/p)fa,p, where fa,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), ga,p lies in the unit ball of HΨ1. Let C be the (finite) norm of jμ. Using a classical computation which gives that |1-〈z,a〉|≤2(1-|a|) for any z∈𝒮(a/|a|,1-|a|), we get |fa,p(z)|≥1/4N/p for such z, hence
(2.21)1≥∫BN¯Ψ2(|ga,p|C)dμ≥μ(S(a|a|,1-|a|))Ψ2(14N/pCΨ1-1((11-|a|)N/p))
for any a∈𝔹N. We may assume that C≥1 and since Ψ2 is concave and vanishes at 0, it follows that
(2.22)μ(S(a|a|,1-|a|))≤4N/pC1Ψ2∘Ψ1-1(1/(1-|a|)N).
(2) We turn to the compactness part. We first prove the sufficient part. Let us assume thatx↦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 ɛ>0, there exists r close enough to 1, such that ∥Ir∥<ɛ (where Ir:HΨ1→LΨ2(𝔹N¯∖r𝔹N,μ)). We fix ɛ>0 and let f be in the unit ball of HΨ2. 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
(2.23)Kμ(h)≲ɛ1/hNΨ2∘Ψ1-1(1/hN),
for h small enough. Then, we apply Lemma 2.8 with η=ɛ and E=𝔹N¯∖r𝔹N to get the existence of r close enough to 1 (independent of f), such that
(2.24)‖Ir(f)‖Ψ2≲μ(BN¯∖rBN)+ɛ∫SNΨ1(Nf)dσ≲μ(BN¯∖rBN)+ɛ‖f‖Ψ1,
for any ɛ>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(𝕊N)=0. Hence
(2.25)limsup⁡r→1‖Ir(f)‖Ψ2≲ɛ‖f‖Ψ1,
for any ɛ>0, which gives (2).
For the converse, we assume thatjμ is compact but that μ is not a vanishing (Ψ1,Ψ2)-Carleson measure, that is there exist ɛ0>0, a sequence (hn)n decreasing to 0 and a sequence (ζn)n⊂𝕊N such that
(2.26)μ(S(ζn,hn))≥ɛ01Ψ2∘Ψ1-1(1/hnN).
Now, let gan,p=Φ-1((1/(1-|an|))N/p)fan,p be as in the proof of the necessity in the boundedness part with an=(1-hn)ζn and p such that Ψ1 is growth function of order p. Since (gan,p)n is bounded in HΨ1 and converges uniformly on every compact subset of 𝔹N, then we must have ∥jμ(gan,p)∥Ψ2→n→∞0, because of Proposition 2.6. But, for z∈𝒮(ζn,hn), we saw that
(2.27)|gan,p(z)|≥14N/pΨ1-1(1hnN).
Since Ψ2 is a growth function of order, let say q, we have, for any t∈(0,1), Ψ2(y/t)≥Ψ2(y)/tq. Hence, we get
(2.28)∫BN¯Ψ2(4Nq/p|gan,p|ɛ01/q)dμ≥Ψ2(1ɛ01/qΨ1-1(1/hnN))μ(S(ζn,hn))≥Ψ2∘Ψ1-1(1/hnN)ɛ0ɛ01Ψ2∘Ψ1-1(1/hnN)=1.
Therefore, ∥jμ(gan,p)∥Ψ2≥ɛ01/q/4Nq/p, which is a contradiction.


#### 2.2.2. Proof of Theorem2.2

We introduce the following function:(2.29)Kμ,α(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN+α+1.
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 constantC̃>0 such that, for every f continuous on 𝔹N and every positive finite Borel measure μ on 𝔹N, one has
(2.30)μ({z∈BN,|z|>1-h,|f(z)|>t})≤C̃Kμ,α(2h)vα({Λf>t}),
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 Lemma2.8 and [11, Lemma 2.4], so we omit it.Lemma 2.10.
Letμ be a finite positive Borel measure on 𝔹N and let Ψ1 and Ψ2 be two subharmonic-preserving growth functions. Assume that there exist A>0, η>0, and hA∈(0,1/2) such that
(2.31)Kμ,α(h)≤η1/hN+1+αψ2(AΨ1-1(1/hN+1+α)),
for every h∈(0,hA). Then, for every f∈AαΨ1(𝔹N) with ∥f∥AαΨ1≤1 and every Borel subset E of 𝔹N, one has
(2.32)∫EΨ2(|f|)dμ≲μ(E)+η∫BNΨ1(Λf)dvα,
where ≲ involves constants which are independent of f, η, and E.We now prove Theorem2.2.Proof of Theorem2.2.
LetΨ1 and Ψ2 be two subharmonic-preserving growth functions such that x↦Ψ2(x)/Ψ1(x) is nondecreasing.
(1) We first assume thatμ 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=𝔹N, gives
(2.33)∫BNΨ2(|f|)dμ≲μ(E)+∫BNΨ1(Λf)dvα.
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 Theorem2.1, using the test function ga,α,p=Φ-1((1/(1-|a|))(N+α+1)/p)fa,α,p, where fa,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 Theorem2.2 is still similar to that for Hardy-Orlicz spaces: we apply Lemma 2.10 to f∈AαΨ1, with η=ɛ and E=𝔹N∖r𝔹N to show that ∥f.χ𝔹N∖r𝔹N∥Ψ2 tends to 0, as r tends to 1. Proposition 2.6 then shows that AαΨ1↪LΨ2(μ) is compact.
For the converse, we procede as for Theorem2.1 using the test function introduced in the boundedness part above.


## 2.1. Statements of the Results

Forζ∈𝔹N¯ and h∈]0,1], we define the nonisotropic “ball’’ of 𝔹N by(2.1)S(ζ,h)={z∈BN,d(ζ,z)2<h}
and its analogue in 𝔹N¯ by(2.2)S(ζ,h)={z∈BN¯,d(ζ,z)2<h}.
Let us also denote by(2.3)Q(ζ,h)=S(ζ,h)∩SN,
the “true’’ balls in 𝕊N. We have σ(Q(ζ,h))≈hN and vα(S(ζ,h))≈hN+α+1 [22].Letμ be a positive Borel measure on 𝔹N¯ whose restriction to 𝕊N 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))≤ChNq/p, while it is a vanishing (q/p)-Carleson measure if μ(𝒮(ζ,h))=o(hNq/p) when h goes to 0. A variant of the well-known Carleson theorem for Hardy spaces [1, 3] ensures that the embedding Hp↪Lq(𝔹N¯,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 Aαp↪Lq(𝔹N,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 thanH1 (resp., Aα1) (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 thanH1 (resp., Aα1), we state that the characterizations of the boundedness and compactness of HΨ1↪LΨ2(𝔹N¯,μ) (resp., AαΨ1↪LΨ2(𝔹N,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 ifΨi(x)=xpi, Ψi=Φpi (with Φ an Orlicz function), or Ψi(x)=xpilogqi(C+x) with p2≥p1, then x↦Ψ2∘Ψ1-1(x)/x is nondecreasing.Theorem 2.1.
LetΨ1 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 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ). Then,(1)
HΨ1 embeds into LΨ2(𝔹N¯,μ) if and only if there exists some h0∈(0,1) such that, for any h∈(0,h0),
(2.4)μ(S(ζ,h))≲1Ψ2∘Ψ1-1(1/hN)
uniformly in ζ∈𝕊N,(2)
the embeddingHΨ1↪LΨ2(𝔹N¯,μ) is compact if and only if
(2.5)μ(S(ζ,h))=oh→0(1Ψ2∘Ψ1-1(1/hN))
uniformly in ζ∈𝕊N.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.
LetΨ1 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 𝔹N. Then,(1)
AαΨ1 embeds into LΨ2(𝔹N,μ) if and only if there exists h0∈(0,1) such that, for any h∈(0,h0),
(2.6)μ(S(ζ,h))≲1Ψ2∘Ψ1-1(1/hN+α+1),
uniformly in ζ∈𝕊N,(2)
the embeddingAαΨ1↪LΨ2(𝔹N,μ) is compact if and only if
(2.7)μ(S(ζ,h))≤oh→0(1Ψ2∘Ψ1-1(1/hN+α+1)),
uniformly in ζ∈𝕊N.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.
Letα>-1 and let 0<p≤q<1 and Ψ1=Φp, Ψ2=Φq as in (3) of Example 1.3. Let μ be a finite positive Borel measure on 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ) (resp., on 𝔹N). Then,(1)
HΦp (resp., AαΦp) embeds into LΦq(𝔹N¯,μ) (resp., LΦq(𝔹N,μ)) if and only if μ is a (q/p)-Carleson measure (resp., a (α,(q/p))-Bergman-Carleson measure),(2)
the embeddingHΦp↪LΦq(𝔹N¯,μ) (resp., AαΦp↪LΦq(𝔹N,μ)) is compact if and only if μ is a vanishing (q/p)-Carleson measure (resp., a vanishing (α,(q/p)) Bergman-Carleson measure).IfΨ1=Ψ2 (or equivalently if Ψ1 and Ψ2 are equivalent), we have the following.Corollary 2.5.
Letα>-1, and let Ψ be subharmonic-preserving growth function. Let μ be a finite positive Borel measure on 𝔹N¯ (whose restriction to 𝕊N is absolutely continuous with respect to σ) (resp., on 𝔹N). Then,(1)
HΨ (resp., AαΨ) embeds into LΨ(𝔹N¯,μ) (resp., LΨ(𝔹N,μ)) if and only if μ is a Carleson measure (resp., a α-Bergman-Carleson measure),(2)
the embeddingHΨ↪LΨ(𝔹N¯,μ) (resp., AαΨ↪LΨ(𝔹N,μ)) is compact if and only if μ is a vanishing Carleson measure (resp., a vanishing α-Bergman-Carleson measure).


## 2.2. Proofs of Theorems2.1 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 HΨ (resp., AαΨ), for Ψ  a subharmonic-preserving growth function, implies the convergence on every compact subset of 𝔹N, but it stems from Proposition 1.10 (resp., Proposition 1.6).Proposition 2.6.
Letα>-1, let Ψ1 and Ψ2 be two subharmonic-preserving growth functions and let μ (resp., μ) be a finite positive Borel measure on 𝔹N¯ (resp., 𝔹N) whose restriction to 𝕊N is absolutely continuous with respect to σ. One assume that jμ:HΨ1↪LΨ2(𝔹N¯,μ) (resp., jμα:AαΨ1↪LΨ2(𝔹N,μ)) is well defined (hence bounded).(1)
The two following assertions are equivalent:
(a)
the canonical embeddingjμ (resp., jμα) is compact;(b)
every sequence in the unit ball ofHΨ1 (resp., AαΨ1), which is convergent to 0 uniformly on every compact subset of 𝔹N, is convergent to 0 in LΨ2(μ) (resp., LΨ2(μ)).(2)
Iflimr→1-∥Ir∥=0 (resp., limr→1-∥Ir,α∥=0), where Ir(f)=f·χ𝔹N¯∖r𝔹N (resp., Ir(f)=f·χ𝔹N∖r𝔹N), then the canonical embedding jμ (resp., jμα) is compact.


### 2.2.1. Proof of Theorem2.1

We assume that the hypothesis of Theorem2.1 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 Kμ, associated to μ by(2.8)Kμ(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN,forh∈(0,1),
so that μ is a Carleson measure if and only if Kμ(h) is finite for some h∈(0,1).Theorem 2.7 (see [12, Theorem 2.4]).
There exist two constantsC̃>0 and C>1 such that, for every f continuous on 𝔹N which admits boundary values almost everywhere on 𝕊N, and every finite positive Borel measure μ on 𝔹N¯, one has
(2.9)μ({z∈BN¯,|z|>1-h,|f(z)|>t})≤C̃Kμ(Ch)σ({Nf>t}),
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.
Letμ be a finite positive Borel measure on 𝔹N¯ 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 hA∈(0,1/C) such that
(2.10)Kμ(h)≤η1/hNΨ2(AΨ1-1(1/hN)),
for every h∈(0,hA). Then, for every f∈HΨ1(𝔹N) such that ∥f∥Ψ1≤1 and every Borel subset E of 𝔹N¯,
(2.11)∫EΨ2(|f|)dμ≲μ(E)+η∫SNΨ1(Nf)dσ,
where ≲ involves a constant which is independent of f, η, and E.Proof.
Letf∈HΨ1(𝔹N) 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. 											(2.12)∫EΨ2(A2N+1(C+1)CN-1|f|)dμ≤∫0xAΨ2′(s)μ(E)ds+ηC̃CN∫xA∞Ψ2′(s)Ψ1((((C+1)CN-1)/A)S)Ψ2(((C+1)/C)S)×σ({Nf>2N+1(C+1)CN-1As})ds,
for some constant xA which depends only on A. Now, since Ψ1 and Ψ2 are two concave growth functions, we have
(2.13)Ψi(s)≲Ψi(Ks)≲Ψi(s),
where ≲ involve constants which only depend on K and Ψi. Therefore, (2.12) becomes
(2.14)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ2′(s)Ψ1(s)Ψ2(s)σ({Nf>2N+1(C+1)CN-1As})ds.
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,
(2.15)Ψ1′(s)Ψ2′(s)≤Ψ1(s)Ψ2(s).
It follows that
(2.16)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ1′(s)σ({Nf>2N+1(C+1)CN-1As})ds≲μ(E)+η∫0+∞Ψ1({ANf2N+1(C+1)CN-1>s})ds=μ(E)+η∫SNψ1(ANf2N+1(C+1)CN-1)dσ≲μ(E)+η∫SNψ1(Nf)dσ.We now can prove Theorem2.1.Proof of Theorem2.1.
(1) We assume that μ is a (Ψ1,Ψ2)-Carleson measure, and we intend to show that jμ:HΨ1↪LΨ2(𝔹N¯,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),
(2.17)μ(S(ζ,h))hN≲1/hNΨ2∘Ψ1-1(1/hN),
uniformly in ζ∈𝕊N. 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),
(2.18)Kμ(h)≲1/hNΨ2∘Ψ1-1(1/hN).
Therefore, we may and will apply Lemma 2.8 with E=𝔹N¯ to get
(2.19)∫BN¯Ψ2(|f|)dμ≲μ(E)+∫SNΨ1(Nf)dσ.
We finish the proof of (1) using Theorem 1.12 and Inequality (1.7):
(2.20)∫BN¯Ψ2(|f|)dμ≲μ(E)+‖f‖Ψ1.
For the converse, letga,p=Φ-1((1/(1-|a|))N/p)fa,p, where fa,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), ga,p lies in the unit ball of HΨ1. Let C be the (finite) norm of jμ. Using a classical computation which gives that |1-〈z,a〉|≤2(1-|a|) for any z∈𝒮(a/|a|,1-|a|), we get |fa,p(z)|≥1/4N/p for such z, hence
(2.21)1≥∫BN¯Ψ2(|ga,p|C)dμ≥μ(S(a|a|,1-|a|))Ψ2(14N/pCΨ1-1((11-|a|)N/p))
for any a∈𝔹N. We may assume that C≥1 and since Ψ2 is concave and vanishes at 0, it follows that
(2.22)μ(S(a|a|,1-|a|))≤4N/pC1Ψ2∘Ψ1-1(1/(1-|a|)N).
(2) We turn to the compactness part. We first prove the sufficient part. Let us assume thatx↦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 ɛ>0, there exists r close enough to 1, such that ∥Ir∥<ɛ (where Ir:HΨ1→LΨ2(𝔹N¯∖r𝔹N,μ)). We fix ɛ>0 and let f be in the unit ball of HΨ2. 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
(2.23)Kμ(h)≲ɛ1/hNΨ2∘Ψ1-1(1/hN),
for h small enough. Then, we apply Lemma 2.8 with η=ɛ and E=𝔹N¯∖r𝔹N to get the existence of r close enough to 1 (independent of f), such that
(2.24)‖Ir(f)‖Ψ2≲μ(BN¯∖rBN)+ɛ∫SNΨ1(Nf)dσ≲μ(BN¯∖rBN)+ɛ‖f‖Ψ1,
for any ɛ>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(𝕊N)=0. Hence
(2.25)limsup⁡r→1‖Ir(f)‖Ψ2≲ɛ‖f‖Ψ1,
for any ɛ>0, which gives (2).
For the converse, we assume thatjμ is compact but that μ is not a vanishing (Ψ1,Ψ2)-Carleson measure, that is there exist ɛ0>0, a sequence (hn)n decreasing to 0 and a sequence (ζn)n⊂𝕊N such that
(2.26)μ(S(ζn,hn))≥ɛ01Ψ2∘Ψ1-1(1/hnN).
Now, let gan,p=Φ-1((1/(1-|an|))N/p)fan,p be as in the proof of the necessity in the boundedness part with an=(1-hn)ζn and p such that Ψ1 is growth function of order p. Since (gan,p)n is bounded in HΨ1 and converges uniformly on every compact subset of 𝔹N, then we must have ∥jμ(gan,p)∥Ψ2→n→∞0, because of Proposition 2.6. But, for z∈𝒮(ζn,hn), we saw that
(2.27)|gan,p(z)|≥14N/pΨ1-1(1hnN).
Since Ψ2 is a growth function of order, let say q, we have, for any t∈(0,1), Ψ2(y/t)≥Ψ2(y)/tq. Hence, we get
(2.28)∫BN¯Ψ2(4Nq/p|gan,p|ɛ01/q)dμ≥Ψ2(1ɛ01/qΨ1-1(1/hnN))μ(S(ζn,hn))≥Ψ2∘Ψ1-1(1/hnN)ɛ0ɛ01Ψ2∘Ψ1-1(1/hnN)=1.
Therefore, ∥jμ(gan,p)∥Ψ2≥ɛ01/q/4Nq/p, which is a contradiction.


### 2.2.2. Proof of Theorem2.2

We introduce the following function:(2.29)Kμ,α(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN+α+1.
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 constantC̃>0 such that, for every f continuous on 𝔹N and every positive finite Borel measure μ on 𝔹N, one has
(2.30)μ({z∈BN,|z|>1-h,|f(z)|>t})≤C̃Kμ,α(2h)vα({Λf>t}),
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 Lemma2.8 and [11, Lemma 2.4], so we omit it.Lemma 2.10.
Letμ be a finite positive Borel measure on 𝔹N and let Ψ1 and Ψ2 be two subharmonic-preserving growth functions. Assume that there exist A>0, η>0, and hA∈(0,1/2) such that
(2.31)Kμ,α(h)≤η1/hN+1+αψ2(AΨ1-1(1/hN+1+α)),
for every h∈(0,hA). Then, for every f∈AαΨ1(𝔹N) with ∥f∥AαΨ1≤1 and every Borel subset E of 𝔹N, one has
(2.32)∫EΨ2(|f|)dμ≲μ(E)+η∫BNΨ1(Λf)dvα,
where ≲ involves constants which are independent of f, η, and E.We now prove Theorem2.2.Proof of Theorem2.2.
LetΨ1 and Ψ2 be two subharmonic-preserving growth functions such that x↦Ψ2(x)/Ψ1(x) is nondecreasing.
(1) We first assume thatμ 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=𝔹N, gives
(2.33)∫BNΨ2(|f|)dμ≲μ(E)+∫BNΨ1(Λf)dvα.
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 Theorem2.1, using the test function ga,α,p=Φ-1((1/(1-|a|))(N+α+1)/p)fa,α,p, where fa,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 Theorem2.2 is still similar to that for Hardy-Orlicz spaces: we apply Lemma 2.10 to f∈AαΨ1, with η=ɛ and E=𝔹N∖r𝔹N to show that ∥f.χ𝔹N∖r𝔹N∥Ψ2 tends to 0, as r tends to 1. Proposition 2.6 then shows that AαΨ1↪LΨ2(μ) is compact.
For the converse, we procede as for Theorem2.1 using the test function introduced in the boundedness part above.


## 2.2.1. Proof of Theorem2.1

We assume that the hypothesis of Theorem2.1 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 Kμ, associated to μ by(2.8)Kμ(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN,forh∈(0,1),
so that μ is a Carleson measure if and only if Kμ(h) is finite for some h∈(0,1).Theorem 2.7 (see [12, Theorem 2.4]).
There exist two constantsC̃>0 and C>1 such that, for every f continuous on 𝔹N which admits boundary values almost everywhere on 𝕊N, and every finite positive Borel measure μ on 𝔹N¯, one has
(2.9)μ({z∈BN¯,|z|>1-h,|f(z)|>t})≤C̃Kμ(Ch)σ({Nf>t}),
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.
Letμ be a finite positive Borel measure on 𝔹N¯ 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 hA∈(0,1/C) such that
(2.10)Kμ(h)≤η1/hNΨ2(AΨ1-1(1/hN)),
for every h∈(0,hA). Then, for every f∈HΨ1(𝔹N) such that ∥f∥Ψ1≤1 and every Borel subset E of 𝔹N¯,
(2.11)∫EΨ2(|f|)dμ≲μ(E)+η∫SNΨ1(Nf)dσ,
where ≲ involves a constant which is independent of f, η, and E.Proof.
Letf∈HΨ1(𝔹N) 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. 											(2.12)∫EΨ2(A2N+1(C+1)CN-1|f|)dμ≤∫0xAΨ2′(s)μ(E)ds+ηC̃CN∫xA∞Ψ2′(s)Ψ1((((C+1)CN-1)/A)S)Ψ2(((C+1)/C)S)×σ({Nf>2N+1(C+1)CN-1As})ds,
for some constant xA which depends only on A. Now, since Ψ1 and Ψ2 are two concave growth functions, we have
(2.13)Ψi(s)≲Ψi(Ks)≲Ψi(s),
where ≲ involve constants which only depend on K and Ψi. Therefore, (2.12) becomes
(2.14)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ2′(s)Ψ1(s)Ψ2(s)σ({Nf>2N+1(C+1)CN-1As})ds.
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,
(2.15)Ψ1′(s)Ψ2′(s)≤Ψ1(s)Ψ2(s).
It follows that
(2.16)∫EΨ2(|f|)dμ≲μ(E)+η∫xA∞Ψ1′(s)σ({Nf>2N+1(C+1)CN-1As})ds≲μ(E)+η∫0+∞Ψ1({ANf2N+1(C+1)CN-1>s})ds=μ(E)+η∫SNψ1(ANf2N+1(C+1)CN-1)dσ≲μ(E)+η∫SNψ1(Nf)dσ.We now can prove Theorem2.1.Proof of Theorem2.1.
(1) We assume that μ is a (Ψ1,Ψ2)-Carleson measure, and we intend to show that jμ:HΨ1↪LΨ2(𝔹N¯,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),
(2.17)μ(S(ζ,h))hN≲1/hNΨ2∘Ψ1-1(1/hN),
uniformly in ζ∈𝕊N. 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),
(2.18)Kμ(h)≲1/hNΨ2∘Ψ1-1(1/hN).
Therefore, we may and will apply Lemma 2.8 with E=𝔹N¯ to get
(2.19)∫BN¯Ψ2(|f|)dμ≲μ(E)+∫SNΨ1(Nf)dσ.
We finish the proof of (1) using Theorem 1.12 and Inequality (1.7):
(2.20)∫BN¯Ψ2(|f|)dμ≲μ(E)+‖f‖Ψ1.
For the converse, letga,p=Φ-1((1/(1-|a|))N/p)fa,p, where fa,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), ga,p lies in the unit ball of HΨ1. Let C be the (finite) norm of jμ. Using a classical computation which gives that |1-〈z,a〉|≤2(1-|a|) for any z∈𝒮(a/|a|,1-|a|), we get |fa,p(z)|≥1/4N/p for such z, hence
(2.21)1≥∫BN¯Ψ2(|ga,p|C)dμ≥μ(S(a|a|,1-|a|))Ψ2(14N/pCΨ1-1((11-|a|)N/p))
for any a∈𝔹N. We may assume that C≥1 and since Ψ2 is concave and vanishes at 0, it follows that
(2.22)μ(S(a|a|,1-|a|))≤4N/pC1Ψ2∘Ψ1-1(1/(1-|a|)N).
(2) We turn to the compactness part. We first prove the sufficient part. Let us assume thatx↦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 ɛ>0, there exists r close enough to 1, such that ∥Ir∥<ɛ (where Ir:HΨ1→LΨ2(𝔹N¯∖r𝔹N,μ)). We fix ɛ>0 and let f be in the unit ball of HΨ2. 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
(2.23)Kμ(h)≲ɛ1/hNΨ2∘Ψ1-1(1/hN),
for h small enough. Then, we apply Lemma 2.8 with η=ɛ and E=𝔹N¯∖r𝔹N to get the existence of r close enough to 1 (independent of f), such that
(2.24)‖Ir(f)‖Ψ2≲μ(BN¯∖rBN)+ɛ∫SNΨ1(Nf)dσ≲μ(BN¯∖rBN)+ɛ‖f‖Ψ1,
for any ɛ>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(𝕊N)=0. Hence
(2.25)limsup⁡r→1‖Ir(f)‖Ψ2≲ɛ‖f‖Ψ1,
for any ɛ>0, which gives (2).
For the converse, we assume thatjμ is compact but that μ is not a vanishing (Ψ1,Ψ2)-Carleson measure, that is there exist ɛ0>0, a sequence (hn)n decreasing to 0 and a sequence (ζn)n⊂𝕊N such that
(2.26)μ(S(ζn,hn))≥ɛ01Ψ2∘Ψ1-1(1/hnN).
Now, let gan,p=Φ-1((1/(1-|an|))N/p)fan,p be as in the proof of the necessity in the boundedness part with an=(1-hn)ζn and p such that Ψ1 is growth function of order p. Since (gan,p)n is bounded in HΨ1 and converges uniformly on every compact subset of 𝔹N, then we must have ∥jμ(gan,p)∥Ψ2→n→∞0, because of Proposition 2.6. But, for z∈𝒮(ζn,hn), we saw that
(2.27)|gan,p(z)|≥14N/pΨ1-1(1hnN).
Since Ψ2 is a growth function of order, let say q, we have, for any t∈(0,1), Ψ2(y/t)≥Ψ2(y)/tq. Hence, we get
(2.28)∫BN¯Ψ2(4Nq/p|gan,p|ɛ01/q)dμ≥Ψ2(1ɛ01/qΨ1-1(1/hnN))μ(S(ζn,hn))≥Ψ2∘Ψ1-1(1/hnN)ɛ0ɛ01Ψ2∘Ψ1-1(1/hnN)=1.
Therefore, ∥jμ(gan,p)∥Ψ2≥ɛ01/q/4Nq/p, which is a contradiction.


## 2.2.2. Proof of Theorem2.2

We introduce the following function:(2.29)Kμ,α(h)=sup⁡0<t<hsup⁡ζ∈SN⁡μ(S(ζ,t))tN+α+1.
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 constantC̃>0 such that, for every f continuous on 𝔹N and every positive finite Borel measure μ on 𝔹N, one has
(2.30)μ({z∈BN,|z|>1-h,|f(z)|>t})≤C̃Kμ,α(2h)vα({Λf>t}),
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 Lemma2.8 and [11, Lemma 2.4], so we omit it.Lemma 2.10.
Letμ be a finite positive Borel measure on 𝔹N and let Ψ1 and Ψ2 be two subharmonic-preserving growth functions. Assume that there exist A>0, η>0, and hA∈(0,1/2) such that
(2.31)Kμ,α(h)≤η1/hN+1+αψ2(AΨ1-1(1/hN+1+α)),
for every h∈(0,hA). Then, for every f∈AαΨ1(𝔹N) with ∥f∥AαΨ1≤1 and every Borel subset E of 𝔹N, one has
(2.32)∫EΨ2(|f|)dμ≲μ(E)+η∫BNΨ1(Λf)dvα,
where ≲ involves constants which are independent of f, η, and E.We now prove Theorem2.2.Proof of Theorem2.2.
LetΨ1 and Ψ2 be two subharmonic-preserving growth functions such that x↦Ψ2(x)/Ψ1(x) is nondecreasing.
(1) We first assume thatμ 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=𝔹N, gives
(2.33)∫BNΨ2(|f|)dμ≲μ(E)+∫BNΨ1(Λf)dvα.
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 Theorem2.1, using the test function ga,α,p=Φ-1((1/(1-|a|))(N+α+1)/p)fa,α,p, where fa,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 Theorem2.2 is still similar to that for Hardy-Orlicz spaces: we apply Lemma 2.10 to f∈AαΨ1, with η=ɛ and E=𝔹N∖r𝔹N to show that ∥f.χ𝔹N∖r𝔹N∥Ψ2 tends to 0, as r tends to 1. Proposition 2.6 then shows that AαΨ1↪LΨ2(μ) is compact.
For the converse, we procede as for Theorem2.1 using the test function introduced in the boundedness part above.


## 3. Applications to Composition Operators

Theorems2.1 and 2.2 will be used to characterize both boundedness and compactness of composition operators on HΨ and AαΨ. As usual, for ϕ:𝔹N→𝔹N holomorphic, Cϕ will be seen as an embedding operator. To do so, we define the pullback measures μϕ and μϕ,α of, respectively, σ and vα under ϕ:(3.1)μϕ(E)=σ(ϕ*-1(E)∩SN),
where ϕ* is the radial limit almost everywhere of ϕ and E is any Borel subset of 𝔹N¯. And(3.2)μϕ,α(E)=vα(ϕ-1(E))
for any Borel subset E of 𝔹N.By a classical formula for pull-back measures,(3.3)‖jμϕ(f)‖Ψ=‖Cϕ(f)‖Ψ(resp.,‖jμϕ,α(f)‖α,Ψ=‖Cϕ(f)‖α,Ψ)
for any growth function Ψ and every polynomial f (resp., for any f∈AαΨ). Then we extend this equality to the whole space HΨ by density of polynomials in HΨ (Theorem 1.13) and using Cauchy’s formula.Remark 3.1.
Observe that it is not true thatjμϕ(f)=Cϕ(f) for any f∈HΨ but it is for any polynomial f. Such an equality holds for every function in AαΨ.Moreover, we have the following criterion for compactness of composition operators.Proposition 3.2.
Letα>-1 and let Ψ1 and Ψ2 be as in Proposition 2.6. Let also ϕ:𝔹N→𝔹N be holomorphic. Cϕ is compact from HΨ1 into HΨ2 (resp., AαΨ1 into AαΨ2) if and only if, for every bounded sequence (fn)n⊂HΨ1 (resp., (fn)n⊂AαΨ1) which converges to 0 on every compact of 𝔹N,∥Cϕ(f)∥Ψ2 (resp., ∥Cϕ(f)∥Ψ2,α) tends to 0.The proof of this proposition is quite similar to that of Proposition2.6.Therefore, Theorems2.1 and 2.2 together with Propositions 2.6 and 3.2 give the following characterizations of the boundedness and compactness of Cϕ on HΨ and AαΨ.Theorem 3.3.
LetΨ1 and Ψ2 be two subharmonic-preserving growth functions such that x↦Ψ2(x)/Ψ1(x) is nondecreasing. Let also ϕ:𝔹N→𝔹N be holomorphic. Then,(1)
Cϕ is bounded from HΨ1 into HΨ2 if and only if μϕ is a (Ψ1,Ψ2)-Carleson measure,(2)
Cϕ is compact from HΨ1 into HΨ2 if and only if μϕ is a vanishing (Ψ1,Ψ2)-Carleson measure.For Bergman-Orlicz spaces, we have the following.Theorem 3.4.
Letα>-1 and let Ψ1 and Ψ2 be two subharmonic-preserving growth functions such that x↦Ψ2(x)/Ψ1(x) is nondecreasing. Let also ϕ:𝔹N→𝔹N be holomorphic. Then,(1)
Cϕ is bounded from AαΨ1 into AαΨ2 if and only if μϕ,α is a (α,Ψ1,Ψ2)-Bergman Carleson measure,(2)
Cϕ is compact from AαΨ1 into AαΨ2 if and only if μϕ,α is a vanishing (α,Ψ1,Ψ2)-Bergman Carleson measure.We immediately deduce the following corollary, which reminds a similar result when the functionΨ is an Orlicz function satisfying the Δ2-Condition (see [11, 12]).Corollary 3.5.
Letα>-1 and let Ψ be a subharmonic preserving growth function. Let also ϕ:𝔹N→𝔹N be holomorphic. Then,(1)
Cϕ is bounded (resp., compact) on HΨ if and only if it is bounded (resp., compact) on one (or equivalently every) Hp, if and only if μϕ is a Carleson measure (resp., a vanishing Carleson measure),(2)
Cϕ is bounded (resp., compact) on AαΨ if and only if it is bounded (resp., compact) on one (or equivalently every) Aαp, if and only if μϕ 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.
Letα>-1, let 0<p≤q<∞ and Ψ1=Φp, and Ψ2=Φq as in (3) of Example 1.3. Let also ϕ:𝔹N→𝔹N be holomorphic. Then,(1)
Cϕ is bounded (resp., compact) from HΨ1 to HΨ2 if and only if it is bounded (resp., compact) from Hp to Hq, if and only if μϕ is a q/p-Carleson measure (resp., a vanishing q/p-Carleson measure),(2)
Cϕ is bounded (resp., compact) from AαΨ1 to AαΨ2 if and only if it is bounded (resp., compact) from Aαp to Aαq, if and only if μϕ is a q/p-Bergman-Carleson measure (resp., a vanishing q/p-Bergman-Carleson measure).



---

*Source: 792763-2012-03-08.xml*