# p-Carleson Measures for a Class of Hardy-Orlicz Spaces

Authors: Benoît Florent Sehba
Journal: International Journal of Mathematics and Mathematical Sciences
 (2011)

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## Abstract

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


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## Body

## 1. Introduction

Hardy-Orlicz spaces are the generalization of the usual Hardy spaces. We raise the question of characterizing those positive measuresμ defined on the unit ball 𝔹n of ℂn such that these spaces embed continuously into the Lebesgue spaces Lp(dμ). More precisely, let denote by dV the Lebesgue measure on 𝔹n and dσ the normalized measure on the unit sphere 𝕊n which is the boundary of 𝔹n. H(𝔹n) denotes the space of holomorphic functions on 𝔹n. Let Φ be continuous and nondecreasing function from [0,∞) onto itself. That is, Φ is a growth function. The Hardy-Orlicz space ℋΦ(𝔹n) is the space of function f in H(𝔹n) such that the functions fr, defined by fr(w)=f(rw) satisfy (1.1)sup⁡r<1inf⁡{λ>0:∫SnΦ(|fr(x)|λ)dσ(x)≤1}<∞.We denote the quantity on the left of the above inequality by∥f∥ℋΦlux or simply ∥f∥ℋΦ when there is no ambiguity. Let us remark that ∥f∥ℋΦlux=supr<1∥fr∥LΦlux, where ∥f∥LΦlux denotes the Luxembourg (quasi)-norm defined by  (1.2)‖f‖LΦlux∶=inf⁡{λ>0:∫SnΦ(|fr(x)|λ)dσ(x)≤1}<∞.Given two growth functionsΦ1 and Φ2, we consider the following question. For which positive measures μ on 𝔹n, the embedding map Iμ:ℋΦ2(𝔹n)→LΦ1(dμ), is continuous? When Φ1 and Φ2 are power functions, such a question has been considered and completely answered in the unit disc and the unit ball in [1–6]. For more general convex growth functions, an attempt to solve the question appears in [7], in the setting of the unit disc where the authors provided with a necessary condition which is not always sufficient and a sufficient condition. The unit ball version of [7] is given in [8]. To be clear at this stage, let us first introduce some usual notations. For any ξ∈𝕊n and δ>0, let (1.3)Bδ(ξ)={w∈Sn:|1-〈w,ξ〉|<δ},Qδ(ξ)={z∈Bn:|1-〈z,ξ〉|<δ}.These are the higher dimension analogues of Carleson regions. We take asΦ1 the power functions, that is, Φ1(t)=tp for 1≤p<∞. Thus, the question is now to characterize those positive measures μ on the unit ball such that there exists a constant C>0 such that  (1.4)∫Bn|f(z)|pdμ(z)≤C(‖f‖HΦlux)p∀f∈HΦ(Bn).
We call such measures p-Carleson measures for ℋΦ(𝔹n). We give a complete answer for a special class of Hardy-Orlicz spaces ℋΦ(𝔹n) with Φ(t)=(t/log(e+t))s, 0<s≤1. For simplicity, we denote this space by ℋs(𝔹n).We prove the following result.Theorem 1.1.
Let0<s≤1 and 1≤p<∞. Then the following assertions are equivalent. (i)
There exists a constantK1>0 such that for any ξ∈𝕊n and δ>0,(1.5)μ(Qδ(ξ))≤K1δn(p/s)(log⁡⁡(4/δ))p.(i)
There exists a constantK2>0 such that(1.6)∫Bn|f(z)|pdμ(z)≤K2‖f‖Hsp∀f∈Hs(Bn).To prove the above result, we combine weak-factorization results for Hardy-Orlicz spaces (see [9, 10]) and some equivalent characterizations of weighted Carleson measures for which we provide an alternative interpretation. We also provide with some further applications of our characterization of the measures considered here to the boundedness of weighted Cesàro-type integral operators from our Hardy-Orlicz spaces to some holomorphic function spaces in Section 3.All over the text,C, Cj and, Kj, j=1,…, will denote positive constant not necessarily the same at each occurrence.This work can be also considered as an application of some recent results obtained by the author and his collaborators [9–11].


## 2.λ-Hardy p-logarithmic Carleson Measures

Forz=(z1,…,zn) and w=(w1,…,wn) in ℂn, we let 〈z,w〉=z1w1¯+⋯+znwn¯ so that |z|2=〈z,z〉=|z1|2+⋯+|zn|2.Recall that whenΦ is a power function, the Hardy-Orlicz space ℋΦ(𝔹n) is just the classical Hardy space. More precisely, for 0<p<∞, let ℋp(𝔹n) denote the Hardy space which is the space of all f∈H(𝔹n) such that (2.1)‖f‖pp∶=sup⁡0<r<1∫Sn|f(rξ)|pdσ(ξ)<∞.
We denote by ℋ∞(𝔹n), the space of bounded analytic functions in 𝔹n.Letρ be a continuous increasing function from [0,∞) onto itself, and such that for some α on [0,1](2.2)ϱ(st)≤sαϱ(t)
for s>1, with st≤1. We define the space BMO(ρ) by (2.3)BMO(ρ)={f∈L2(Sn);sup⁡Binf⁡R∈PN(B)1(ϱ(σ(B)))2σ(B)∫B|f-R|2dσ<∞},
where  for B=Bδ(ξ0), the space 𝒫N(B) is the space of polynomials of order ≤N in the (2n-1) last coordinates related to an orthonormal basis whose first element is ξ0 and second element ℑξ0. The integer N is taken larger than 2nα-1. For C, the quantity appearing in the definition of BMO(ρ), we note ∥f∥BMO(ρ)∶=∥f∥2+C. The space BMOA(ρ) is then the space of function f∈ℋ2(𝔹n) such that (2.4)sup⁡r<1‖fr‖BMO(ρ)<∞.
Clearly, BMOA(ρ) coincides with the space of holomorphic functions in ℋ2(𝔹n) such that their boundary values lie in BMO(ρ). The space BMOA(1) is the usual space of function with bounded mean oscillation BMOA while the space of function of logarithmic mean oscillation LMOA is given by 1/ρ(t)=log4/t.Letμ denote a positive Borel measure on 𝔹n. The measure μ is called an s-Carleson measure, if there is a finite constant C>0 such that for any ξ∈𝕊n and any 0<δ<1, (2.5)μ(Qδ(ξ))≤C(σ(Bδ(ξ)))s.
When s=1, μ is just called Carleson measure. The infinimum of all these constants C will be denoted ∥μ∥s. We use the notation ∥μ∥ for ∥μ∥1. In this section, we are interested in Carleson measure with weights involving the logarithmic function. Let μ be a positive Borel measure on 𝔹n and 0<s<∞. For ρ, a positive function defined on (0,1), we say μ is a (ρ,s)-Carleson measure if there is a constant C>0 such that for any ξ∈𝕊n and 0<δ<1,(2.6)μ(Qδ(ξ))≤C(σ(Bδ(ξ)))sρ(δ).
If s=1, μ is called a ρ-Carleson measure.We will restrict here to the caseρ(t)=(log(4/t))p(loglog(e4/t))q, 0<p,q<∞ studied by the author in [11] (see also [12] for a special case in one dimension). But here we go beyond the interpretation provided in [11].


### 2.1.λ-Hardy ρ-Carleson Measures

In this section, we recall some results of [11] and the notion of λ-Hardy Carleson measures. We then provide with an alternative interpretation of the results of [11] that will be useful to our characterization. From now on, the notation K1≈K2, where K1 and K2 are two positive constants, will mean there exists an absolute positive constant M such that (2.7)M-1K2≤K1≤MK2,
and in this case, we say K1 and K2 are comparable or equivalent. The notation K1≲K2 means K1≤MK2 for some absolute positive constant M. Let set (2.8)Ka(z)=(1-|a|2)n|1-〈a,z〉|2n.We first recall the following higher dimension version of the theorem of Carleson [1] and its reproducing kernel formulation.Theorem 2.1.
For a positive Borel measureμ on 𝔹n, and 0<p<∞, the following are equivalent (i)
The measureμ is a Carleson measure.(ii)
There is a constantK1>0 such that, for all f∈ℋp(𝔹n),
(2.9)∫Bn|f(z)|pdμ(z)≤K1‖f‖pp.(iii)
There is a constantK2>0 such that, for all a∈𝔹n,
(2.10)∫BnKa(w)dμ(w)≤K2<∞.We note that the constantsK1, K2 in Theorem 2.1 are both comparable to ∥μ∥. The proof of this theorem can be found in [13].We now recall some basic facts aboutλ-Hardy measures.Definition 2.2.
Let0<p, q<∞ and λ=q/p. We say a positive measure μ on 𝔹n is a λ-Hardy Carleson measure if there exists a constant C>0 such that for all f∈ℋp(𝔹n),
(2.11)∫Bn|f(z)|qdμ(z)≤C‖f‖Hpq.The following high dimension Peter Duren's characterization ofλ-Hardy Carleson measures is useful for our purpose.Proposition 2.3.
Let0<p, q<∞ and λ=q/p>1. Let μ be a positive measure on 𝔹n. Then the following assertions are equivalent.(i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.12)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))λ.(ii)
There exists a constantK2>0 such that
(2.13)sup⁡a∈Bn∫BnKaλ(z)dμ(z)<K2<∞.(iii)
There exists a constantK3>0 such that for all f∈ℋp(𝔹n),
(2.14)∫Bn|f(z)|qdμ(z)≤K3‖f‖Hpq.The constantsK1, K2, and K3 in the above proposition are equivalent. That (i)⇔(ii) can be found in [11]. The equivalence (i)⇔(iii) can be found in [14] for example. We have the following elementary consequence.Corollary 2.4.
Let0≤p, q<∞, p≠0 and let μ be a positive measure on 𝔹n. Then the following assertion are equivalent. (i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.15)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))1+(q/p).(ii)
There exists a constantK2>0 such that
(2.16)sup⁡a∈Bn∫BnKa1+(q/p)(z)dμ(z)≤K2<∞.(iii)
There exists a constantK3>0 such that for all f∈ℋp(𝔹n),
(2.17)sup⁡a∈Bn∫BnKa(z)|f(z)|qdμ(z)≤K3∥f∥Hpq.(iv)
There exists a constantK4>0 such that for all f∈ℋp(𝔹n) and any g∈ℋr(𝔹n),
(2.18)∫Bn|f(z)|q|g(z)|rdμ(z)≤K4‖f‖Hpq‖g‖Hrr.Proof.
The equivalence (i)⇔(ii) is a special case of Proposition 2.3. Note that (iii) is equivalent in saying that for any f∈ℋp(𝔹n), the measure (|f(z)|qdμ(z))/∥f∥ℋpq is a Carleson measure which is equivalent to (iv). The implication (iv)⇒(i) follows from the usual arguments. Thus, it only remains to prove that (ii)⇒(iii). First by Proposition 2.3, (ii) is equivalent in saying that there exists a constant K2′>0 such that for any f∈ℋp(𝔹n),
(2.19)∫Bn|f(z)|p+qdμ(z)≤K2′‖f‖Hpp+q.
It follows from the hypotheses, the latter, and Hölder's inequality that
(2.20)∫BnKa(z)|f(z)|qdμ(z)≤(∫BnKa(z)1+(q/p)dμ(z))p/(p+q)(∫Bn|f(z)|p+qdμ(z))q/(p+q),≤K2K2′‖f‖Hpq.
Thus (ii)⇒(iii). The proof is complete.Next, we recall the following result proved in [11].Theorem 2.5.
Let0≤p, q<∞, s≥1, and let μ be a positive Borel measure on 𝔹n. Then the following conditions are equivalent.(i)
There isK1>0 such that for any ξ∈𝕊n and 0<δ<1,
(2.21)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))s(log⁡⁡(4/δ))p(loglog(e4/δ))q.(ii)
There isK2>0 such that
(2.22)sup⁡a∈Bn(log⁡⁡41-|a|)p(logloge41-|a|)q∫BnKa(z)sdμ(z)≤K2<∞.(iii)
There isK3>0 such that for any f∈BMOA,
(2.23)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)s|f(z)|pdμ(z)≤K3‖f‖BMOAp.(iv)
There isK4>0 such that for any g∈LMOA,
(2.24)sup⁡a∈Bn(log⁡⁡41-|a|)q∫BnKa(z)s|g(z)|qdμ(z)≤K‖g‖LMOAq.(v)
There isK5>0 such that for any f∈BMOA and any g∈LMOA,
(2.25)sup⁡a∈Bn∫BnKa(z)s|f(z)|p|g(z)|qdμ(z)≤K5‖f‖BMOAp‖f‖LMOAq.Definition 2.6.
Let0<p, q<∞ and λ=q/p. Let ρ be a positive function defined on [0,∞). We say a positive measure μ on 𝔹n is a λ-Hardy ρ-Carleson measure if for any f∈ℋp(𝔹n), the measure
(2.26)dμ̃(z)=|f(z)|q‖f‖pqdμ(z)
is a ρ-Carleson measure.We have the following characterization ofλ-Hardy ρ-Carleson measure which is in fact an alternative interpretation of Theorem 2.5.Theorem 2.7.
Let0≤p,q,r,s<∞, s≠0, and let μ be a positive Borel measure on 𝔹n. Then the following conditions are equivalent.(i)
There isK1>0 such that for any ξ∈𝕊n and 0<δ<1,
(2.27)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))1+(r/s)(log⁡⁡(4/δ))p(loglog(e4/δ))q.(ii)
There isK2>0 such that for any f∈BMOA, and any h∈ℋs(𝔹n),
(2.28)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)|h(z)|r|f(z)|pdμ(z)≤K2‖h‖Hsr‖f‖BMOAp.(iii)
There isK3>0 such that for any g∈LMOA, and anyh∈ℋs(𝔹n),
(2.29)sup⁡a∈Bn(log⁡⁡41-|a|)p∫BnKa(z)|h(z)|r|g(z)|qdμ(z)≤K2‖h‖Hsr‖g‖LMOAq.(iv)
There isK4>0 such that for any f∈BMO, any g∈LMOA, and any h∈ℋs(𝔹n),
(2.30)sup⁡a∈Bn∫BnKa(z)|h(z)|r|f(z)|p|g(z)|qdμ(z)≤K3‖h‖Hsr‖f‖BMOAp‖g‖LMOAq.(v)
There isK5>0 such that for any f∈BMOA, g∈LMOA, and any h∈ℋs(𝔹n) and l∈ℋm(𝔹n),
(2.31)∫Bn|f(z)|p|g(z)|q|h(z)|r|l(z)|mdμ(z)≤K5‖h‖Hsr‖l‖Hmm‖f‖BMOAp‖g‖LMOAq.Proof.
(i)⇔(iv): we first observe with Theorem 2.5 that (i) is equivalent in saying that there is a constant C1 such that for any f∈BMOA and any g∈LMOA,
(2.32)sup⁡a∈Bn∫BnKa(z)1+(r/s)|f(z)|p|g(z)|qdμ(z)≤C1‖f‖BMOAp‖g‖LMOAq.
By Corollary 2.4, the latter is equivalent to (iv).
(ii)⇔(iii)⇔(iv): by rewriting (ii) as
(2.33)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)|f(z)|pdμ̃(z)≤K2‖h‖Hsr‖f‖BMOAp,
where dμ̃(z)=(|h(z)|r/∥h∥ℋsr)dμ(z), it follows directly from Theorem 2.5 that (ii)⇔(iii)⇔(iv).
That (iv)⇔(v) is a consequence of Theorem 2.1. The proof is complete.


### 2.2.p-Carleson Measures for Hardy-Orlicz Spaces

In this section, we characterizep-Carleson measures of some special Hardy-Orlicz spaces. For this, we will need a weak factorization result of functions in these spaces which follows from the one in [10].Proposition 2.8.
Let0<s≤1. Let ℋs(𝔹n) denote the Hardy-Orlicz space corresponding to the function Φ(t)=(t/log(e+t))s. Then the following assertions hold. (i)
The product of two functions, one inℋs(𝔹n) and the other one in BMOA, is in ℋs(𝔹n). Moreover,
(2.34)∥fg∥Hs≲∥f∥Hs∥g∥BMOA.(ii)
Any functionf in the unit ball of ℋs(𝔹n) admits the following representation (weak factorization):
(2.35)f=∑jfjgj,fj∈Hs(Bn),gj∈BMOA
with
(2.36)∑j=0∞‖fj‖Hs‖gj‖BMOA≲‖f‖Hs.Let us remark that the spaceℋ1(𝔹n) is the predual of LMOA. The following theorem gives a characterization of p-Carleson measures of the Hardy-Orlicz spaces considered here.Theorem 2.9.
Let0<s≤1, 1≤p<∞. Let ℋs(𝔹n) be the Hardy-Orlicz space ℋΦ(𝔹n) corresponding to the function Φ(t)=(t/log(e+t))s. Then, for μ a positive measure on 𝔹n, the following assertions are equivalent. (i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.37)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))(p/s)(log⁡⁡(4/δ))p.(ii)
There exists a constantK2>0 such that for any f∈ℋs(𝔹n),
(2.38)∫Bn|f(z)|pdμ(z)≤K2‖f‖Hsp.Proof.
We remark that if (2.38) holds in the unit ball of ℋs(𝔹n), then it holds for all f∈ℋs(𝔹n). Recall that by Proposition 2.8, every function f in the unit ball of ℋs(𝔹n) weakly factorizes as
(2.39)f=∑j=0∞fjgj
and ∑j=0∞∥fj∥ℋs∥gj∥BMOA≲∥f∥ℋs. It follows using the equivalent assertion (iv) of Theorem 2.7 that
(2.40)(∫Bn|f(z)|pdμ(z))1/p=(∫Bn|∑j=0∞fj(z)gj(z)|pdμ(z))1/p≤∑j=0∞(∫Bn|fj(z)gj(z)|pdμ(z))1/p=∑j=0∞(∫Bn|fj(z)|s|fj(z)|p-s|gj(z)|pdμ(z))1/p≲∑j=0∞(‖fj‖Hss‖fj‖Hsp-s‖gj‖BMOAp)1/p=∑j=0∞‖fj‖Hs‖gj‖BMOA≲‖f‖Hs.
Now we prove that (ii)⇒(i). That (ii) holds implies in particular that for any f∈ℋs(𝔹n) and any g∈BMOA,
(2.41)∫Bn|f(z)|p|g(z)|pdμ(z)≤K2‖f‖Hsp‖g‖BMOAp.
We observe with Corollary 2.4 that (2.41) is equivalent in saying that for any g∈BMOA, the measure
(2.42)dμ̃(z)=|g(z)|p‖g‖BMOApdμ(z)
is a (p/s)-Carleson measure or equivalently,
(2.43)sup⁡a∈Bn∫BnKa(z)(p/s)|g(z)|pdμ(z)≤K3‖g‖BMOAp.
By Theorem 2.5, the latter is equivalent to
(2.44)sup⁡a∈Bn(log⁡⁡41-|a|)p∫BnKa(z)(p/s)dμ(z)≤K4,
which is equivalent to (i). The proof is complete.


## 2.1.λ-Hardy ρ-Carleson Measures

In this section, we recall some results of [11] and the notion of λ-Hardy Carleson measures. We then provide with an alternative interpretation of the results of [11] that will be useful to our characterization. From now on, the notation K1≈K2, where K1 and K2 are two positive constants, will mean there exists an absolute positive constant M such that (2.7)M-1K2≤K1≤MK2,
and in this case, we say K1 and K2 are comparable or equivalent. The notation K1≲K2 means K1≤MK2 for some absolute positive constant M. Let set (2.8)Ka(z)=(1-|a|2)n|1-〈a,z〉|2n.We first recall the following higher dimension version of the theorem of Carleson [1] and its reproducing kernel formulation.Theorem 2.1.
For a positive Borel measureμ on 𝔹n, and 0<p<∞, the following are equivalent (i)
The measureμ is a Carleson measure.(ii)
There is a constantK1>0 such that, for all f∈ℋp(𝔹n),
(2.9)∫Bn|f(z)|pdμ(z)≤K1‖f‖pp.(iii)
There is a constantK2>0 such that, for all a∈𝔹n,
(2.10)∫BnKa(w)dμ(w)≤K2<∞.We note that the constantsK1, K2 in Theorem 2.1 are both comparable to ∥μ∥. The proof of this theorem can be found in [13].We now recall some basic facts aboutλ-Hardy measures.Definition 2.2.
Let0<p, q<∞ and λ=q/p. We say a positive measure μ on 𝔹n is a λ-Hardy Carleson measure if there exists a constant C>0 such that for all f∈ℋp(𝔹n),
(2.11)∫Bn|f(z)|qdμ(z)≤C‖f‖Hpq.The following high dimension Peter Duren's characterization ofλ-Hardy Carleson measures is useful for our purpose.Proposition 2.3.
Let0<p, q<∞ and λ=q/p>1. Let μ be a positive measure on 𝔹n. Then the following assertions are equivalent.(i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.12)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))λ.(ii)
There exists a constantK2>0 such that
(2.13)sup⁡a∈Bn∫BnKaλ(z)dμ(z)<K2<∞.(iii)
There exists a constantK3>0 such that for all f∈ℋp(𝔹n),
(2.14)∫Bn|f(z)|qdμ(z)≤K3‖f‖Hpq.The constantsK1, K2, and K3 in the above proposition are equivalent. That (i)⇔(ii) can be found in [11]. The equivalence (i)⇔(iii) can be found in [14] for example. We have the following elementary consequence.Corollary 2.4.
Let0≤p, q<∞, p≠0 and let μ be a positive measure on 𝔹n. Then the following assertion are equivalent. (i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.15)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))1+(q/p).(ii)
There exists a constantK2>0 such that
(2.16)sup⁡a∈Bn∫BnKa1+(q/p)(z)dμ(z)≤K2<∞.(iii)
There exists a constantK3>0 such that for all f∈ℋp(𝔹n),
(2.17)sup⁡a∈Bn∫BnKa(z)|f(z)|qdμ(z)≤K3∥f∥Hpq.(iv)
There exists a constantK4>0 such that for all f∈ℋp(𝔹n) and any g∈ℋr(𝔹n),
(2.18)∫Bn|f(z)|q|g(z)|rdμ(z)≤K4‖f‖Hpq‖g‖Hrr.Proof.
The equivalence (i)⇔(ii) is a special case of Proposition 2.3. Note that (iii) is equivalent in saying that for any f∈ℋp(𝔹n), the measure (|f(z)|qdμ(z))/∥f∥ℋpq is a Carleson measure which is equivalent to (iv). The implication (iv)⇒(i) follows from the usual arguments. Thus, it only remains to prove that (ii)⇒(iii). First by Proposition 2.3, (ii) is equivalent in saying that there exists a constant K2′>0 such that for any f∈ℋp(𝔹n),
(2.19)∫Bn|f(z)|p+qdμ(z)≤K2′‖f‖Hpp+q.
It follows from the hypotheses, the latter, and Hölder's inequality that
(2.20)∫BnKa(z)|f(z)|qdμ(z)≤(∫BnKa(z)1+(q/p)dμ(z))p/(p+q)(∫Bn|f(z)|p+qdμ(z))q/(p+q),≤K2K2′‖f‖Hpq.
Thus (ii)⇒(iii). The proof is complete.Next, we recall the following result proved in [11].Theorem 2.5.
Let0≤p, q<∞, s≥1, and let μ be a positive Borel measure on 𝔹n. Then the following conditions are equivalent.(i)
There isK1>0 such that for any ξ∈𝕊n and 0<δ<1,
(2.21)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))s(log⁡⁡(4/δ))p(loglog(e4/δ))q.(ii)
There isK2>0 such that
(2.22)sup⁡a∈Bn(log⁡⁡41-|a|)p(logloge41-|a|)q∫BnKa(z)sdμ(z)≤K2<∞.(iii)
There isK3>0 such that for any f∈BMOA,
(2.23)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)s|f(z)|pdμ(z)≤K3‖f‖BMOAp.(iv)
There isK4>0 such that for any g∈LMOA,
(2.24)sup⁡a∈Bn(log⁡⁡41-|a|)q∫BnKa(z)s|g(z)|qdμ(z)≤K‖g‖LMOAq.(v)
There isK5>0 such that for any f∈BMOA and any g∈LMOA,
(2.25)sup⁡a∈Bn∫BnKa(z)s|f(z)|p|g(z)|qdμ(z)≤K5‖f‖BMOAp‖f‖LMOAq.Definition 2.6.
Let0<p, q<∞ and λ=q/p. Let ρ be a positive function defined on [0,∞). We say a positive measure μ on 𝔹n is a λ-Hardy ρ-Carleson measure if for any f∈ℋp(𝔹n), the measure
(2.26)dμ̃(z)=|f(z)|q‖f‖pqdμ(z)
is a ρ-Carleson measure.We have the following characterization ofλ-Hardy ρ-Carleson measure which is in fact an alternative interpretation of Theorem 2.5.Theorem 2.7.
Let0≤p,q,r,s<∞, s≠0, and let μ be a positive Borel measure on 𝔹n. Then the following conditions are equivalent.(i)
There isK1>0 such that for any ξ∈𝕊n and 0<δ<1,
(2.27)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))1+(r/s)(log⁡⁡(4/δ))p(loglog(e4/δ))q.(ii)
There isK2>0 such that for any f∈BMOA, and any h∈ℋs(𝔹n),
(2.28)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)|h(z)|r|f(z)|pdμ(z)≤K2‖h‖Hsr‖f‖BMOAp.(iii)
There isK3>0 such that for any g∈LMOA, and anyh∈ℋs(𝔹n),
(2.29)sup⁡a∈Bn(log⁡⁡41-|a|)p∫BnKa(z)|h(z)|r|g(z)|qdμ(z)≤K2‖h‖Hsr‖g‖LMOAq.(iv)
There isK4>0 such that for any f∈BMO, any g∈LMOA, and any h∈ℋs(𝔹n),
(2.30)sup⁡a∈Bn∫BnKa(z)|h(z)|r|f(z)|p|g(z)|qdμ(z)≤K3‖h‖Hsr‖f‖BMOAp‖g‖LMOAq.(v)
There isK5>0 such that for any f∈BMOA, g∈LMOA, and any h∈ℋs(𝔹n) and l∈ℋm(𝔹n),
(2.31)∫Bn|f(z)|p|g(z)|q|h(z)|r|l(z)|mdμ(z)≤K5‖h‖Hsr‖l‖Hmm‖f‖BMOAp‖g‖LMOAq.Proof.
(i)⇔(iv): we first observe with Theorem 2.5 that (i) is equivalent in saying that there is a constant C1 such that for any f∈BMOA and any g∈LMOA,
(2.32)sup⁡a∈Bn∫BnKa(z)1+(r/s)|f(z)|p|g(z)|qdμ(z)≤C1‖f‖BMOAp‖g‖LMOAq.
By Corollary 2.4, the latter is equivalent to (iv).
(ii)⇔(iii)⇔(iv): by rewriting (ii) as
(2.33)sup⁡a∈Bn(logloge41-|a|)q∫BnKa(z)|f(z)|pdμ̃(z)≤K2‖h‖Hsr‖f‖BMOAp,
where dμ̃(z)=(|h(z)|r/∥h∥ℋsr)dμ(z), it follows directly from Theorem 2.5 that (ii)⇔(iii)⇔(iv).
That (iv)⇔(v) is a consequence of Theorem 2.1. The proof is complete.


## 2.2.p-Carleson Measures for Hardy-Orlicz Spaces

In this section, we characterizep-Carleson measures of some special Hardy-Orlicz spaces. For this, we will need a weak factorization result of functions in these spaces which follows from the one in [10].Proposition 2.8.
Let0<s≤1. Let ℋs(𝔹n) denote the Hardy-Orlicz space corresponding to the function Φ(t)=(t/log(e+t))s. Then the following assertions hold. (i)
The product of two functions, one inℋs(𝔹n) and the other one in BMOA, is in ℋs(𝔹n). Moreover,
(2.34)∥fg∥Hs≲∥f∥Hs∥g∥BMOA.(ii)
Any functionf in the unit ball of ℋs(𝔹n) admits the following representation (weak factorization):
(2.35)f=∑jfjgj,fj∈Hs(Bn),gj∈BMOA
with
(2.36)∑j=0∞‖fj‖Hs‖gj‖BMOA≲‖f‖Hs.Let us remark that the spaceℋ1(𝔹n) is the predual of LMOA. The following theorem gives a characterization of p-Carleson measures of the Hardy-Orlicz spaces considered here.Theorem 2.9.
Let0<s≤1, 1≤p<∞. Let ℋs(𝔹n) be the Hardy-Orlicz space ℋΦ(𝔹n) corresponding to the function Φ(t)=(t/log(e+t))s. Then, for μ a positive measure on 𝔹n, the following assertions are equivalent. (i)
There exists a constantK1>0 such that for any ξ∈𝕊n and any 0<δ<1,
(2.37)μ(Qδ(ξ))≤K1(σ(Bδ(ξ)))(p/s)(log⁡⁡(4/δ))p.(ii)
There exists a constantK2>0 such that for any f∈ℋs(𝔹n),
(2.38)∫Bn|f(z)|pdμ(z)≤K2‖f‖Hsp.Proof.
We remark that if (2.38) holds in the unit ball of ℋs(𝔹n), then it holds for all f∈ℋs(𝔹n). Recall that by Proposition 2.8, every function f in the unit ball of ℋs(𝔹n) weakly factorizes as
(2.39)f=∑j=0∞fjgj
and ∑j=0∞∥fj∥ℋs∥gj∥BMOA≲∥f∥ℋs. It follows using the equivalent assertion (iv) of Theorem 2.7 that
(2.40)(∫Bn|f(z)|pdμ(z))1/p=(∫Bn|∑j=0∞fj(z)gj(z)|pdμ(z))1/p≤∑j=0∞(∫Bn|fj(z)gj(z)|pdμ(z))1/p=∑j=0∞(∫Bn|fj(z)|s|fj(z)|p-s|gj(z)|pdμ(z))1/p≲∑j=0∞(‖fj‖Hss‖fj‖Hsp-s‖gj‖BMOAp)1/p=∑j=0∞‖fj‖Hs‖gj‖BMOA≲‖f‖Hs.
Now we prove that (ii)⇒(i). That (ii) holds implies in particular that for any f∈ℋs(𝔹n) and any g∈BMOA,
(2.41)∫Bn|f(z)|p|g(z)|pdμ(z)≤K2‖f‖Hsp‖g‖BMOAp.
We observe with Corollary 2.4 that (2.41) is equivalent in saying that for any g∈BMOA, the measure
(2.42)dμ̃(z)=|g(z)|p‖g‖BMOApdμ(z)
is a (p/s)-Carleson measure or equivalently,
(2.43)sup⁡a∈Bn∫BnKa(z)(p/s)|g(z)|pdμ(z)≤K3‖g‖BMOAp.
By Theorem 2.5, the latter is equivalent to
(2.44)sup⁡a∈Bn(log⁡⁡41-|a|)p∫BnKa(z)(p/s)dμ(z)≤K4,
which is equivalent to (i). The proof is complete.


## 3. Some Applications

We provide in this section with some applications ofp-Carleson measures of the above Hardy-Orlicz spaces to the boundedness of multiplication operators, composition operators, and Cesàro integral-type operators. Let us first introduce the generalized Bergman spaces in the unit ball. We recall that for f∈H(𝔹n), its radial derivative Rf is the holomorphic function defined by (3.1)Rf(z)=∑j=1nzj∂f∂zj(z).
Let α∈ℝ, 1≤p<∞ with α+p>-1. The generalized Bergman space 𝒜αp(𝔹n) consists of holomorphic function f such that (3.2)‖f‖p,αp∶=∫Bn|Rf(z)|p(1-|z|2)α+pdV(z)<∞.
Clearly, 𝒜αp(𝔹n) is a Banach under  (3.3)‖f‖p,αp∶=|f(0)|+∫Bn|Rf(z)|p(1-|z|2)α+pdV(z)<∞.
These spaces have been studied in [15]. When α>-1, the space 𝒜αp(𝔹n) corresponds to the usual weighted Bergman space which consists of holomorphic function f in 𝔹n such that (3.4)‖f‖p,αp∶=∫Bn|f(z)|p(1-|z|2)αdV(z)<∞.
For α=-1 and p=2, the corresponding space is just the Hardy space ℋ2(𝔹n).Letu be a holomorphic function in 𝔹n. We denote by ℳu the multiplication operator by u defined on H(𝔹n) by (3.5)Mu(f)(z)=u(z)f(z),f∈H(Bn).
We recall that if φ is a holomorphic self map of 𝔹n, then the composition operator Cφ is defined on H(𝔹n) by (3.6)Cφ(f)∶=f∘φ.
For u a holomorphic function in 𝔹n, the weighted composition operator uCφ is the composition operator followed by the multiplication by u. That is, (3.7)uCφ(f)=Mu(f∘φ)=u(f∘φ).
For b a holomorphic function in 𝔹n, the Cesàro-type integral operator Tb is defined by (3.8)Tb(f)(z)=∫01f(tz)Rg(tz)dtt,g,f∈H(Bn).
Combining this operator with the weighted composition operator, we obtain a more general operator Tu,φ,b=Tb(ℳu(f∘φ))=Tb(u(f∘φ)) given by (3.9)Tu,φ,b(f)(z)=∫01u(tz)(f∘φ)(tz)Rg(tz)dtt,f∈H(Bn).
When φ(z)=z for all z∈𝔹n, we write Tu,φ,b=Tu,b. The multiplication operator, the composition operator, the Cesàro-type integral, and their products have been intensively studied by many authors on various holomorphic function spaces. We refer to the following and the references therein [11, 12, 16–30]. As an application of the characterization of p-Carleson measures for the Hardy-Orlicz spaces of the previous section, we consider boundedness criteria of the above operators from Hardy-Orlicz spaces to (generalized) weighted Bergman spaces and weighted BMOA spaces in the unit ball. We have the following result.Theorem 3.1.
Let0<s≤1, 1≤p<∞ and, α>-1. Then uCφ is bounded from ℋs(𝔹n) to 𝒜αp(𝔹n) if and only if
(3.10)sup⁡a∈Bn(log⁡⁡4(1-|a|))p∫Bn((1-|a|2)n|1-〈φ(z),a〉|2n)(p/s)|u(z)|p(1-|z|2)αdV(z)<∞.Proof.
Clearly, thatuCφ is bounded from ℋs(𝔹n) to 𝒜αp(𝔹n) is equivalent in saying that there is a constant C>0 such that for any f∈ℋs(𝔹n),
(3.11)∫Bn|f∘φ(z)|p|u(z)|p(1-|z|2)αdV(z)≤C‖f‖Hsp.
Let us write dVα(z)=(1-|z|2)αdV(z), dVα,u(z)=|u(z)|pdVα(z). If μ=Vα,u∘φ-1, then an easy change of variables gives that (3.11) is equivalent to
(3.12)∫Bn|f(z)|pdμ(z)≤C‖f‖Hsp.
The latter inequality is equivalent in saying that the measure μ is a p-Carleson measure for ℋs(𝔹n). It follows from Theorem 2.9 and the equivalent definitions in Theorem 2.7 that (3.11) is equivalent to
(3.13)sup⁡a∈Bn(log⁡⁡4(1-|a|))p∫Bn((1-|a|2)n|1-〈w,a〉|2n)p/sdμ(w)<∞.
Changing the variables back, we finally obtain that uCφ is bounded from ℋΦs(𝔹n) to 𝒜αp(𝔹n) if and only if
(3.14)sup⁡a∈Bn(log⁡⁡4(1-|a|))p∫Bn((1-|a|2)n|1-〈φ(z),a〉|2n)(p/s)|u(z)|p(1-|z|2)αdV(z)<∞.
The proof is complete.Remarking that one has(3.15)R(Tbf)(z)=f(z)Rb(z)foranyg,f∈H(Bn),
we prove in the same way the following result.Theorem 3.2.
Let0<s≤1,1≤p<∞ and α∈ℝ with α+p>-1. Then Tu,φ,b is bounded from ℋs(𝔹n) to 𝒜αp(𝔹n) if and only if
(3.16)sup⁡a∈Bn(log⁡⁡41-|a|)p∫Bn((1-|a|2)n|1-〈φ(z),a〉|2n)p/sdμ(z)<∞,
where dμ(z)=|u(z)|p|Rb(z)|p(1-|z|2)α+pdV(z).
Let us consider now the operatorTu,b. We have the following:Theorem 3.3.
Let0<s≤1, 0≤p, q<∞, and α>-1. Let 1/ρ(t)=(log(4/t))p(loglog(e4/t))q. Then Tu,b is bounded from ℋs(𝔹n) to BMOA(ρ), if and only if
(3.17)sup⁡a∈Bn(log⁡⁡41-|a|)2(p+1)(logloge41-|a|)2q∫Bn((1-|a|2)n|1-〈z,a〉|2n)1+(2/s)dμ(z)<∞,
with dμ(z)=|u(z)|2|Rb(z)|2(1-|z|2)dV(z).Proof.
We recall that a functionh is in BMOA(ρ) if and only if the measure |Rh(z)|2(1-|z|2)dV(z) is a (1/ρ2)-Carleson measure (see [31]). That is
(3.18)sup⁡a∈Bn(log⁡⁡41-|a|)2p(logloge41-|a|)2q∫Bn(1-|a|2)n|1-〈z,a〉|2n|Rh(z)|2(1-|z|2)dV(z)<∞.
It follows that Tu,b is bounded from ℋs(𝔹n) to BMOA(ρ) if and only if for any f∈ℋs(𝔹n),
(3.19)sup⁡a∈Bn(log⁡⁡41-|a|)2p(logloge41-|a|)2q∫Bn(1-|a|2)n|1-〈z,a〉|2n|f(z)|2dμ(z)≤C‖f‖Hs2,dμ(z)=|u(z)|2|Rb(z)|2(1-|z|2)dV(z). By the equivalent definition in Theorem 2.7, this is equivalent in saying that for any f1∈BMOA, f2∈LMOA, and any g∈ℋm(𝔹n),
(3.20)∫Bn|f(z)|2|g(z)|m|f1(z)|2p|f2(z)|2qdμ(z)≤C‖f‖Hs2‖f1‖BMOA2p‖f2‖LMOA2q‖g‖mm,
which is equivalent in saying that the measure
(3.21)dμ̃(z)=|f1(z)|2p|f2(z)|2q‖f1‖BMOA2p‖f2‖LMOA2q|g(z)|m‖g‖mm|u(z)|2|Rb(z)|2(1-|z|2)dV(z)
is a 2-Carleson measure for ℋs(𝔹n). It follows from the equivalent definitions of Theorems 2.7 and 2.9 that the latter is equivalent to
(3.22)sup⁡a∈Bn(log⁡⁡41-|a|)2(p+1)(logloge41-|a|)2q∫Bn((1-|a|2)n|1-〈z,a〉|2n)1+(2/s)dμ(z)<∞.
The proof is complete.The methods used in this text are quite specific to the case considered here, that is, the embeddingIμ:ℋs(𝔹n)→Lp(𝔹n). We remark that even in the case 0<s≤p<1, the condition (i) of Theorem 2.9 is still necessary. The proof given here does not allow to say if it is sufficient. In general, the characterization of those positive measures μ on 𝔹n such that the embedding map Iμ:ℋΦ1(𝔹n)→ℋΦ2(𝔹n) (Φ1≠Φ2 if Φ1 and Φ2 are convex growth functions) is bounded, is still open.



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