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

Abstract

An alternative interpretation of a family of weighted Carleson measures is used to characterize 𝑝-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.

1. Introduction

Hardy-Orlicz spaces are the generalization of the usual Hardy spaces. We raise the question of characterizing those positive measures 𝜇 defined on the unit ball 𝔹^𝑛 of ℂ^𝑛 such that these spaces embed continuously into the Lebesgue spaces 𝐿^𝑝(𝑑𝜇). More precisely, let denote by 𝑑𝑉 the Lebesgue measure on 𝔹^𝑛 and 𝑑𝜎 the normalized measure on the unit sphere 𝕊^𝑛 which is the boundary of 𝔹^𝑛. 𝐻(𝔹^𝑛) denotes the space of holomorphic functions on 𝔹^𝑛. Let Φ be continuous and nondecreasing function from [0,∞) onto itself. That is, Φ is a growth function. The Hardy-Orlicz space ℋ^Φ(𝔹^𝑛) is the space of function 𝑓 in 𝐻(𝔹^𝑛) such that the functions 𝑓_𝑟, defined by 𝑓_𝑟(𝑤)=𝑓(𝑟𝑤) satisfy sup_𝑟<1inf𝜆>0∶_𝕊𝑛Φ𝑓_𝑟(𝑥)𝜆𝑑𝜎(𝑥)≤1<∞. (1.1)

We denote the quantity on the left of the above inequality by ‖𝑓‖^lux_ℋΦ or simply ‖𝑓‖_ℋ^Φ when there is no ambiguity. Let us remark that ‖𝑓‖^lux_ℋΦ=sup_𝑟<1‖𝑓_𝑟‖^lux_𝐿Φ, where ‖𝑓‖^lux_𝐿Φ denotes the Luxembourg (quasi)-norm defined by 𝑓^lux_𝐿Φ∶=inf𝜆>0∶_𝕊𝑛Φ𝑓_𝑟(𝑥)𝜆𝑑𝜎(𝑥)≤1<∞. (1.2)

Given two growth functions Φ₁ and Φ₂, we consider the following question. For which positive measures 𝜇 on 𝔹^𝑛, the embedding map 𝐼_𝜇∶ℋ^Φ2(𝔹^𝑛)→𝐿^Φ1(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∈𝕊^𝑛 and 𝛿>0, let 𝐵_𝛿(𝜉)=𝑤∈𝕊^𝑛∶,𝑄1−𝑤,𝜉<𝛿_𝛿(𝜉)=𝑧∈𝔹^𝑛∶1−𝑧,𝜉<𝛿. (1.3)

These are the higher dimension analogues of Carleson regions. We take as Φ₁ the power functions, that is, Φ₁(𝑡)=𝑡^𝑝 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 _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐶𝑓^lux_ℋΦ^𝑝∀𝑓∈ℋ^Φ𝔹^𝑛. (1.4) We call such measures 𝑝-Carleson measures for ℋ^Φ(𝔹^𝑛). We give a complete answer for a special class of Hardy-Orlicz spaces ℋ^Φ(𝔹^𝑛) with Φ(𝑡)=(𝑡/log(𝑒+𝑡))^𝑠, 0<𝑠≤1. For simplicity, we denote this space by ℋ_𝑠(𝔹^𝑛).

We prove the following result.

Theorem 1.1. Let 0<𝑠≤1 and 1≤𝑝<∞. Then the following assertions are equivalent. (i)There exists a constant 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and 𝛿>0, 𝜇𝑄_𝛿(𝜉)≤𝐾₁𝛿^{𝑛(𝑝/𝑠)}log(4/𝛿)^𝑝. (1.5)(i)There exists a constant 𝐾₂>0 such that _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₂𝑓^𝑝_ℋ𝑠∀𝑓∈ℋ_𝑠𝔹^𝑛. (1.6)

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, 𝐶_𝑗 and, 𝐾_𝑗, 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

For 𝑧=(𝑧₁,…,𝑧_𝑛) and 𝑤=(𝑤₁,…,𝑤_𝑛) in ℂ^𝑛, we let ⟨𝑧,𝑤⟩=𝑧₁𝑤₁+⋯+𝑧_𝑛𝑤_𝑛 so that |𝑧|²=⟨𝑧,𝑧⟩=|𝑧₁|²+⋯+|𝑧_𝑛|².

Recall that when Φ is a power function, the Hardy-Orlicz space ℋ^Φ(𝔹^𝑛) is just the classical Hardy space. More precisely, for 0<𝑝<∞, let ℋ^𝑝(𝔹^𝑛) denote the Hardy space which is the space of all 𝑓∈𝐻(𝔹^𝑛) such that 𝑓^𝑝_𝑝∶=sup_{0<𝑟<1𝕊𝑛}𝑓(𝑟𝜉)^𝑝𝑑𝜎(𝜉)<∞. (2.1) We denote by ℋ^∞(𝔹^𝑛), the space of bounded analytic functions in 𝔹^𝑛.

Let 𝜌 be a continuous increasing function from [0,∞) onto itself, and such that for some 𝛼 on [0,1]𝜚(𝑠𝑡)≤𝑠^𝛼𝜚(𝑡) (2.2) for 𝑠>1, with 𝑠𝑡≤1. We define the space BMO(𝜌) by BMO𝜌=𝑓∈𝐿²𝕊^𝑛;sup_𝐵inf_{𝑅∈𝒫𝑁(𝐵)}1𝜚(𝜎(𝐵))²𝜎(𝐵)^𝐵𝑓−𝑅²𝑑𝜎<∞, (2.3) where for 𝐵=𝐵_𝛿(𝜉₀), the space 𝒫_𝑁(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(𝔹^𝑛) such that sup_𝑟<1𝑓_{𝑟BMO(𝜌)}<∞. (2.4) Clearly, BMO𝐴(𝜌) coincides with the space of holomorphic functions in ℋ²(𝔹^𝑛) 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.

Let 𝜇 denote a positive Borel measure on 𝔹^𝑛. The measure 𝜇 is called an 𝑠-Carleson measure, if there is a finite constant 𝐶>0 such that for any 𝜉∈𝕊^𝑛 and any 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤𝐶𝜎𝐵_𝛿(𝜉)^𝑠. (2.5) 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 𝔹^𝑛 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∈𝕊^𝑛 and 0<𝛿<1,𝜇𝑄_𝛿(𝜉)𝜎𝐵≤𝐶^𝛿(𝜉)^𝑠𝜌(𝛿). (2.6) If 𝑠=1, 𝜇 is called a 𝜌-Carleson measure.

We will restrict here to the case 𝜌(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 𝑀⁻¹𝐾₂≤𝐾₁≤𝑀𝐾₂, (2.7) 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 𝐾_𝑎(𝑧)=1−|𝑎|^2𝑛|1−⟨𝑎,𝑧⟩|^2𝑛. (2.8)

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 measure 𝜇 on 𝔹^𝑛, and 0<𝑝<∞, the following are equivalent (i)The measure 𝜇 is a Carleson measure. (ii)There is a constant 𝐾₁>0 such that, for all 𝑓∈ℋ^𝑝(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₁𝑓^𝑝_𝑝. (2.9)(iii)There is a constant 𝐾₂>0 such that, for all 𝑎∈𝔹^𝑛, _𝔹𝑛𝐾_𝑎(𝑤)𝑑𝜇(𝑤)≤𝐾₂<∞. (2.10)

We note that the constants 𝐾₁, 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 about 𝜆-Hardy measures.

Definition 2.2. Let 0<𝑝, 𝑞<∞ and 𝜆=𝑞/𝑝. We say a positive measure 𝜇 on 𝔹^𝑛 is a 𝜆-Hardy Carleson measure if there exists a constant 𝐶>0 such that for all 𝑓∈ℋ^𝑝(𝔹^𝑛), _𝔹n𝑓(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐶𝑓^𝑞_ℋ𝑝. (2.11)

The following high dimension Peter Duren's characterization of 𝜆-Hardy Carleson measures is useful for our purpose.

Proposition 2.3. Let 0<𝑝, 𝑞<∞ and 𝜆=𝑞/𝑝>1. Let 𝜇 be a positive measure on 𝔹^𝑛. Then the following assertions are equivalent.(i)There exists a constant 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and any 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤𝐾₁𝜎𝐵_𝛿(𝜉)^𝜆. (2.12)(ii)There exists a constant 𝐾₂>0 such that sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾^𝜆_𝑎(𝑧)𝑑𝜇(𝑧)<𝐾₂<∞. (2.13)(iii)There exists a constant 𝐾₃>0 such that for all 𝑓∈ℋ^𝑝(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾₃𝑓^𝑞_ℋ𝑝. (2.14)

The constants 𝐾₁, 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. Let 0≤𝑝, 𝑞<∞, 𝑝≠0 and let 𝜇 be a positive measure on 𝔹^𝑛. Then the following assertion are equivalent. (i)There exists a constant 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and any 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤𝐾₁𝜎𝐵_𝛿(𝜉)^{1+(𝑞/𝑝)}. (2.15)(ii)There exists a constant 𝐾₂>0 such that sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾_𝑎^{1+(𝑞/𝑝)}(𝑧)𝑑𝜇(𝑧)≤𝐾₂<∞. (2.16)(iii)There exists a constant 𝐾₃>0 such that for all 𝑓∈ℋ^𝑝(𝔹^𝑛), sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾_𝑎(𝑧)𝑓(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾₃‖𝑓‖^𝑞_ℋ𝑝. (2.17)(iv)There exists a constant 𝐾₄>0 such that for all 𝑓∈ℋ^𝑝(𝔹^𝑛) and any 𝑔∈ℋ^𝑟(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^𝑞𝑔(𝑧)^𝑟𝑑𝜇(𝑧)≤𝐾₄𝑓^𝑞_ℋ𝑝𝑔^𝑟_ℋ𝑟. (2.18)

Proof. The equivalence (i)⇔(ii) is a special case of Proposition 2.3. Note that (iii) is equivalent in saying that for any 𝑓∈ℋ^𝑝(𝔹^𝑛), the measure (|𝑓(𝑧)|^𝑞𝑑𝜇(𝑧))/‖𝑓‖^𝑞_ℋ𝑝 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 𝐾^′₂>0 such that for any 𝑓∈ℋ^𝑝(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^𝑝+𝑞𝑑𝜇(𝑧)≤𝐾^′₂𝑓_ℋ^𝑝+𝑞_𝑝. (2.19) It follows from the hypotheses, the latter, and Hölder's inequality that _𝔹𝑛𝐾_𝑎(𝑧)𝑓(𝑧)^𝑞𝑑𝜇(𝑧)≤_𝔹𝑛𝐾_𝑎(𝑧)^{1+(𝑞/𝑝)}𝑑𝜇(𝑧)^{𝑝/(𝑝+𝑞)}_𝔹𝑛𝑓(𝑧)^𝑝+𝑞𝑑𝜇(𝑧)^{𝑞/(𝑝+𝑞)},≤𝐾₂𝐾^′₂𝑓^𝑞_ℋ𝑝. (2.20) Thus (ii)⇒(iii). The proof is complete.

Next, we recall the following result proved in [11].

Theorem 2.5. Let 0≤𝑝, 𝑞<∞, 𝑠≥1, and let 𝜇 be a positive Borel measure on 𝔹^𝑛. Then the following conditions are equivalent.(i)There is 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤𝐾₁𝜎𝐵_𝛿(𝜉)^𝑠log(4/𝛿)^𝑝loglog𝑒⁴/𝛿^𝑞. (2.21)(ii)There is 𝐾₂>0 such that sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^𝑝loglog𝑒⁴1−|𝑎|^𝑞_𝔹^𝑛𝐾_𝑎(𝑧)^𝑠𝑑𝜇(𝑧)≤𝐾₂<∞. (2.22)(iii)There is 𝐾₃>0 such that for any 𝑓∈𝐵𝑀𝑂𝐴, sup_{𝑎∈𝔹𝑛}loglog𝑒⁴1−|𝑎|^𝑞_𝔹^𝑛𝐾_𝑎(𝑧)^𝑠𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₃𝑓^𝑝_{𝐵𝑀𝑂𝐴}. (2.23)(iv)There is 𝐾₄>0 such that for any 𝑔∈𝐿𝑀𝑂𝐴, sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^𝑞_𝔹^𝑛𝐾_𝑎(𝑧)^𝑠𝑔(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾𝑔^𝑞_{𝐿𝑀𝑂𝐴}. (2.24)(v)There is 𝐾₅>0 such that for any 𝑓∈𝐵𝑀𝑂𝐴 and any 𝑔∈𝐿𝑀𝑂𝐴, sup_{𝑎∈𝔹𝑛𝐵𝑛}𝐾_𝑎(𝑧)^𝑠𝑓(𝑧)^𝑝𝑔(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾₅𝑓^𝑝_{𝐵𝑀𝑂𝐴}^𝑞_{𝐿𝑀𝑂𝐴}. (2.25)

Definition 2.6. Let 0<𝑝, 𝑞<∞ and 𝜆=𝑞/𝑝. Let 𝜌 be a positive function defined on [0,∞). We say a positive measure 𝜇 on 𝔹^𝑛 is a 𝜆-Hardy 𝜌-Carleson measure if for any 𝑓∈ℋ^𝑝(𝔹^𝑛), the measure 𝑑𝜇(𝑧)=𝑓(𝑧)^𝑞𝑓^𝑞_𝑝𝑑𝜇(𝑧) (2.26) is a 𝜌-Carleson measure.

We have the following characterization of 𝜆-Hardy 𝜌-Carleson measure which is in fact an alternative interpretation of Theorem 2.5.

Theorem 2.7. Let 0≤𝑝,𝑞,𝑟,𝑠<∞, 𝑠≠0, and let 𝜇 be a positive Borel measure on 𝔹^𝑛. Then the following conditions are equivalent.(i)There is 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤K₁𝜎𝐵_𝛿(𝜉)^{1+(𝑟/𝑠)}log(4/𝛿)^𝑝loglog𝑒⁴/𝛿^𝑞. (2.27)(ii)There is 𝐾₂>0 such that for any 𝑓∈𝐵𝑀𝑂𝐴, and any ℎ∈ℋ^𝑠(𝔹^𝑛), sup_{𝑎∈𝔹𝑛}loglog𝑒⁴1−|𝑎|^𝑞_𝔹^𝑛𝐾_𝑎(𝑧)|ℎ(𝑧)|^𝑟𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₂‖ℎ‖^𝑟_ℋ^𝑠𝑓^𝑝_{𝐵𝑀𝑂𝐴}. (2.28)(iii)There is 𝐾₃>0 such that for any 𝑔∈𝐿𝑀𝑂𝐴, and anyℎ∈ℋ^𝑠(𝔹^𝑛), sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^𝑝_𝔹^𝑛𝐾_𝑎(𝑧)|ℎ(𝑧)|^𝑟𝑔(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾₂‖ℎ‖^𝑟_ℋ^𝑠𝑔^𝑞_{𝐿𝑀𝑂𝐴}. (2.29)(iv)There is 𝐾₄>0 such that for any 𝑓∈𝐵𝑀𝑂, any 𝑔∈𝐿𝑀𝑂𝐴, and any ℎ∈ℋ^𝑠(𝔹^𝑛), sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾_𝑎(𝑧)|ℎ(𝑧)|^𝑟𝑓(𝑧)^𝑝𝑔(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐾₃‖ℎ‖^𝑟_ℋ^𝑠𝑓^𝑝_{𝐵𝑀𝑂𝐴}𝑔^𝑞_{𝐿𝑀𝑂𝐴}. (2.30)(v)There is 𝐾₅>0 such that for any 𝑓∈𝐵𝑀𝑂𝐴, 𝑔∈𝐿𝑀𝑂𝐴, and any ℎ∈ℋ^𝑠(𝔹^𝑛) and 𝑙∈ℋ^𝑚(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^p𝑔(𝑧)^𝑞|ℎ(𝑧)|^𝑟|𝑙(𝑧)|^𝑚𝑑𝜇(𝑧)≤𝐾₅‖ℎ‖^𝑟_ℋ^𝑠‖𝑙‖^𝑚_ℋ^𝑚𝑓^𝑝_{𝐵𝑀𝑂𝐴}𝑔^𝑞_{𝐿𝑀𝑂𝐴}. (2.31)

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, sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾_𝑎(𝑧)^{1+(𝑟/𝑠)}𝑓(𝑧)^𝑝𝑔(𝑧)^𝑞𝑑𝜇(𝑧)≤𝐶₁𝑓^𝑝_BMO𝐴𝑔^𝑞_LMO𝐴. (2.32) By Corollary 2.4, the latter is equivalent to (iv).
(ii)⇔(iii)⇔(iv): by rewriting (ii) as sup_{𝑎∈𝔹𝑛}loglog𝑒⁴1−|𝑎|^𝑞_𝔹^𝑛𝐾_𝑎(𝑧)𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₂‖ℎ‖^𝑟_ℋ^𝑠𝑓^𝑝_BMO𝐴, (2.33) where 𝑑𝜇(𝑧)=(|ℎ(𝑧)|^𝑟/‖ℎ‖^𝑟_ℋ^𝑠)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 characterize 𝑝-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. Let 0<𝑠≤1. Let ℋ_𝑠(𝔹^𝑛) 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 in ℋ^𝑠(𝔹^𝑛) and the other one in 𝐵𝑀𝑂𝐴, is in ℋ_𝑠(𝔹^𝑛). Moreover, ‖𝑓𝑔‖_ℋ𝑠≲‖𝑓‖_ℋ^𝑠‖𝑔‖_{𝐵𝑀𝑂𝐴}. (2.34)(ii)Any function 𝑓 in the unit ball of ℋ_𝑠(𝔹^𝑛) admits the following representation (weak factorization): 𝑓=_𝑗𝑓_𝑗𝑔_𝑗,𝑓_𝑗∈ℋ^𝑠𝔹^𝑛,𝑔_𝑗∈𝐵𝑀𝑂𝐴 (2.35) with ^∞_𝑗=0𝑓_𝑗ℋ𝑠𝑔_{𝑗𝐵𝑀𝑂𝐴}≲𝑓_ℋ𝑠. (2.36)

Let us remark that the space ℋ₁(𝔹^𝑛) 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. Let 0<𝑠≤1, 1≤𝑝<∞. Let ℋ_𝑠(𝔹^𝑛) be the Hardy-Orlicz space ℋ^Φ(𝔹^𝑛) corresponding to the function Φ(𝑡)=(𝑡/log(𝑒+𝑡))^s. Then, for 𝜇 a positive measure on 𝔹^𝑛, the following assertions are equivalent. (i)There exists a constant 𝐾₁>0 such that for any 𝜉∈𝕊^𝑛 and any 0<𝛿<1, 𝜇𝑄_𝛿(𝜉)≤𝐾₁𝜎𝐵_𝛿(𝜉)^(𝑝/𝑠)log(4/𝛿)^𝑝. (2.37)(ii)There exists a constant 𝐾₂>0 such that for any 𝑓∈ℋ_𝑠(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₂𝑓^𝑝_ℋ𝑠. (2.38)

Proof. We remark that if (2.38) holds in the unit ball of ℋ_𝑠(𝔹^𝑛), then it holds for all 𝑓∈ℋ_𝑠(𝔹^𝑛). Recall that by Proposition 2.8, every function 𝑓 in the unit ball of ℋ_𝑠(𝔹^𝑛) weakly factorizes as 𝑓=^∞_𝑗=0𝑓_𝑗𝑔_𝑗 (2.39) and ∑^∞_𝑗=0‖𝑓_𝑗‖_ℋ^𝑠‖𝑔_𝑗‖_BMO𝐴≲‖𝑓‖_ℋ𝑠. It follows using the equivalent assertion (iv) of Theorem 2.7 that _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)^1/𝑝=_𝔹𝑛^∞_𝑗=0𝑓_𝑗(𝑧)𝑔_𝑗(𝑧)^𝑝𝑑𝜇(𝑧)^1/𝑝≤^∞_{𝑗=0𝔹𝑛}𝑓_𝑗(𝑧)𝑔_𝑗(𝑧)^𝑝𝑑𝜇(𝑧)^1/𝑝=^∞_{𝑗=0𝔹𝑛}𝑓_𝑗(𝑧)^𝑠𝑓_𝑗(𝑧)^𝑝−𝑠𝑔_𝑗(𝑧)^𝑝𝑑𝜇(𝑧)^1/𝑝≲^∞_𝑗=0𝑓_𝑗^𝑠_ℋ𝑠𝑓_𝑗ℋ^𝑝−𝑠_𝑠𝑔_𝑗^𝑝_BMO𝐴^1/𝑝=^∞_𝑗=0𝑓_𝑗ℋ𝑠𝑔_𝑗BMO𝐴≲𝑓_ℋ𝑠. (2.40) Now we prove that (ii)⇒(i). That (ii) holds implies in particular that for any 𝑓∈ℋ^𝑠(𝔹^𝑛) and any 𝑔∈BMO𝐴, _𝔹𝑛𝑓(𝑧)^𝑝𝑔(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₂𝑓^𝑝_ℋ𝑠𝑔^𝑝_BMO𝐴. (2.41) We observe with Corollary 2.4 that (2.41) is equivalent in saying that for any 𝑔∈BMO𝐴, the measure 𝑑𝜇(𝑧)=𝑔(𝑧)^𝑝𝑔^𝑝_BMO𝐴𝑑𝜇(𝑧) (2.42) is a (𝑝/𝑠)-Carleson measure or equivalently, sup_{𝑎∈𝔹𝑛𝔹𝑛}𝐾_𝑎(𝑧)^(𝑝/𝑠)𝑔(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐾₃𝑔^𝑝_BMO𝐴. (2.43) By Theorem 2.5, the latter is equivalent to sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^𝑝_𝔹^𝑛𝐾_𝑎(𝑧)^(𝑝/𝑠)𝑑𝜇(𝑧)≤𝐾₄, (2.44) which is equivalent to (𝑖). The proof is complete.

3. Some Applications

We provide in this section with some applications of 𝑝-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(𝔹^𝑛), its radial derivative 𝑅𝑓 is the holomorphic function defined by 𝑅𝑓(𝑧)=^𝑛_𝑗=1𝑧_𝑗𝜕𝑓𝜕𝑧_𝑗(𝑧). (3.1) Let 𝛼∈ℝ, 1≤𝑝<∞ with 𝛼+𝑝>−1. The generalized Bergman space 𝒜^𝑝_𝛼(𝔹^𝑛) consists of holomorphic function 𝑓 such that 𝑓^𝑝_𝑝,𝛼∶=_𝔹𝑛𝑅𝑓(𝑧)^𝑝1−|𝑧|^2𝛼+𝑝𝑑𝑉(𝑧)<∞. (3.2) Clearly, 𝒜^𝑝_𝛼(𝔹^𝑛) is a Banach under 𝑓^𝑝_𝑝,𝛼∶=𝑓(0)+_𝔹𝑛𝑅𝑓(z)^𝑝1−|𝑧|^2𝛼+𝑝𝑑𝑉(𝑧)<∞. (3.3) These spaces have been studied in [15]. When 𝛼>−1, the space 𝒜^𝑝_𝛼(𝔹^𝑛) corresponds to the usual weighted Bergman space which consists of holomorphic function 𝑓 in 𝔹^𝑛 such that 𝑓^𝑝_𝑝,𝛼∶=_𝔹𝑛𝑓(𝑧)^𝑝1−|𝑧|^2𝛼𝑑𝑉(𝑧)<∞. (3.4) For 𝛼=−1 and 𝑝=2, the corresponding space is just the Hardy space ℋ²(𝔹^𝑛).

Let 𝑢 be a holomorphic function in 𝔹^𝑛. We denote by ℳ_𝑢 the multiplication operator by 𝑢 defined on 𝐻(𝔹^𝑛) by ℳ_𝑢𝑓(𝑧)=𝑢(𝑧)𝑓(𝑧),𝑓∈𝐻𝔹^𝑛. (3.5) We recall that if 𝜑 is a holomorphic self map of 𝔹^𝑛, then the composition operator 𝐶_𝜑 is defined on 𝐻(𝔹^𝑛) by 𝐶_𝜑𝑓∶=𝑓∘𝜑. (3.6) For 𝑢 a holomorphic function in 𝔹^𝑛, the weighted composition operator 𝑢𝐶_𝜑 is the composition operator followed by the multiplication by 𝑢. That is, 𝑢𝐶_𝜑𝑓=ℳ_𝑢𝑓∘𝜑=𝑢𝑓∘𝜑. (3.7) For 𝑏 a holomorphic function in 𝔹^𝑛, the Cesàro-type integral operator 𝑇_𝑏 is defined by 𝑇_𝑏𝑓(𝑧)=¹₀𝑓(𝑡𝑧)𝑅𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑔,𝑓∈H𝔹^𝑛. (3.8) Combining this operator with the weighted composition operator, we obtain a more general operator 𝑇_{𝑢,𝜑,𝑏}=𝑇_𝑏(ℳ_𝑢(𝑓∘𝜑))=𝑇_𝑏(𝑢(𝑓∘𝜑)) given by 𝑇_{𝑢,𝜑,𝑏}𝑓(𝑧)=¹₀𝑢(𝑡𝑧)𝑓∘𝜑(𝑡𝑧)𝑅𝑔(𝑡𝑧)𝑑𝑡𝑡,𝑓∈𝐻𝔹^𝑛. (3.9) When 𝜑(𝑧)=𝑧 for all 𝑧∈𝔹^𝑛, we write 𝑇_{𝑢,𝜑,𝑏}=𝑇_𝑢,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. Let 0<𝑠≤1, 1≤𝑝<∞ and, 𝛼>−1. Then 𝑢𝐶_𝜑 is bounded from ℋ_𝑠(𝔹^𝑛) to 𝒜^𝑝_𝛼(𝔹^𝑛) if and only if sup_{𝑎∈𝔹𝑛}4log(1−|𝑎|)^𝑝_𝔹^𝑛1−|𝑎|^2𝑛1−𝜑(𝑧),𝑎^{2𝑛(𝑝/𝑠)}|𝑢(𝑧)|^𝑝1−|𝑧|^2𝛼𝑑𝑉(𝑧)<∞. (3.10)

Proof. Clearly, that 𝑢𝐶_𝜑 is bounded from ℋ_𝑠(𝔹^𝑛) to 𝒜^𝑝_𝛼(𝔹^𝑛) is equivalent in saying that there is a constant 𝐶>0 such that for any 𝑓∈ℋ_𝑠(𝔹^𝑛), 𝑓∘𝜑(_𝔹𝑛𝑧)^𝑝|𝑢(𝑧)|^𝑝1−|𝑧|^2𝛼𝑑𝑉(𝑧)≤𝐶𝑓^𝑝_ℋ𝑠. (3.11) Let us write 𝑑𝑉_𝛼(𝑧)=(1−|𝑧|²)^𝛼𝑑𝑉(𝑧), 𝑑𝑉_𝛼,𝑢(𝑧)=|𝑢(𝑧)|^𝑝𝑑𝑉_𝛼(𝑧). If 𝜇=𝑉_𝛼,𝑢∘𝜑⁻¹, then an easy change of variables gives that (3.11) is equivalent to _𝔹𝑛𝑓(𝑧)^𝑝𝑑𝜇(𝑧)≤𝐶𝑓^𝑝_ℋ𝑠. (3.12) The latter inequality is equivalent in saying that the measure 𝜇 is a 𝑝-Carleson measure for ℋ_𝑠(𝔹^𝑛). It follows from Theorem 2.9 and the equivalent definitions in Theorem 2.7 that (3.11) is equivalent to sup_{𝑎∈𝔹𝑛}4log(1−|𝑎|)^𝑝_𝔹^𝑛1−|𝑎|^2𝑛|1−⟨𝑤,𝑎⟩|^{2𝑛𝑝/𝑠}𝑑𝜇(𝑤)<∞. (3.13) Changing the variables back, we finally obtain that 𝑢𝐶_𝜑 is bounded from ℋ^Φ𝑠(𝔹^𝑛) to 𝒜^𝑝_𝛼(𝔹^𝑛) if and only if sup_{𝑎∈𝔹𝑛}4log(1−|𝑎|)^𝑝_𝔹^𝑛1−|𝑎|^2𝑛1−𝜑(𝑧),𝑎^{2𝑛(𝑝/𝑠)}|𝑢(𝑧)|^𝑝1−|𝑧|^2𝛼𝑑𝑉(𝑧)<∞. (3.14) The proof is complete.

Remarking that one has 𝑅𝑇_𝑏𝑓(𝑧)=𝑓(𝑧)𝑅𝑏(𝑧)forany𝑔,𝑓∈𝐻𝔹^𝑛, (3.15) we prove in the same way the following result.

Theorem 3.2. Let 0<𝑠≤1,1≤𝑝<∞ and 𝛼∈ℝ with 𝛼+𝑝>−1. Then 𝑇_{𝑢,𝜑,𝑏} is bounded from ℋ_𝑠(𝔹^𝑛) to 𝒜^𝑝_𝛼(𝔹ⁿ) if and only if sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^𝑝_𝔹^𝑛1−|𝑎|^2𝑛1−𝜑(𝑧),𝑎^{2𝑛𝑝/𝑠}𝑑𝜇(𝑧)<∞, (3.16) where 𝑑𝜇(𝑧)=|𝑢(𝑧)|^𝑝|𝑅𝑏(𝑧)|^𝑝(1−|𝑧|²)^𝛼+𝑝𝑑𝑉(𝑧).
Let us consider now the operator 𝑇_𝑢,𝑏. We have the following:

Theorem 3.3. Let 0<𝑠≤1, 0≤𝑝, 𝑞<∞, and 𝛼>−1. Let 1/𝜌(𝑡)=(log(4/𝑡))^𝑝(loglog(𝑒⁴/𝑡))^𝑞. Then 𝑇_𝑢,𝑏 is bounded from ℋ_𝑠(𝔹^𝑛) to 𝐵𝑀𝑂𝐴(𝜌), if and only if sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^2(𝑝+1)loglog𝑒⁴1−|𝑎|^2𝑞_𝔹^𝑛1−|𝑎|^2𝑛|1−⟨𝑧,𝑎⟩|^{2𝑛1+(2/𝑠)}𝑑𝜇(𝑧)<∞, (3.17) with 𝑑𝜇(𝑧)=|𝑢(𝑧)|²|𝑅𝑏(𝑧)|²(1−|𝑧|²)𝑑𝑉(𝑧).

Proof. We recall that a function ℎ 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 sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^2𝑝loglog𝑒⁴1−|𝑎|^2𝑞_𝔹^𝑛1−|𝑎|^2𝑛|1−⟨𝑧,𝑎⟩|^2𝑛|𝑅ℎ(𝑧)|²1−|𝑧|²𝑑𝑉(𝑧)<∞. (3.18) It follows that 𝑇_𝑢,𝑏 is bounded from ℋ_𝑠(𝔹^𝑛) to BMO𝐴(𝜌) if and only if for any 𝑓∈ℋ_𝑠(𝔹^𝑛), sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^2𝑝loglog𝑒⁴1−|𝑎|^2𝑞_𝔹^𝑛1−|𝑎|^2𝑛|1−⟨𝑧,𝑎⟩|^2𝑛𝑓(𝑧)²𝑑𝜇(𝑧)≤𝐶𝑓²_ℋ𝑠, (3.19)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∈ℋ^𝑚(𝔹^𝑛), _𝔹𝑛𝑓(𝑧)²𝑔(𝑧)^𝑚𝑓₁(𝑧)^2𝑝𝑓₂(𝑧)^2𝑞𝑑𝜇(𝑧)≤𝐶𝑓²_ℋ𝑠𝑓₁^2𝑝_BMO𝐴𝑓₂^2𝑞_LMO𝐴𝑔^𝑚_𝑚, (3.20) which is equivalent in saying that the measure 𝑑𝜇(𝑧)=𝑓₁(𝑧)^2𝑝𝑓₂(𝑧)^2𝑞𝑓₁^2𝑝_BMO𝐴𝑓₂^2𝑞_LMO𝐴𝑔(𝑧)^𝑚𝑔^𝑚_𝑚|𝑢(𝑧)|²|𝑅𝑏(𝑧)|²1−|𝑧|²𝑑𝑉(𝑧) (3.21) is a 2-Carleson measure for ℋ_𝑠(𝔹^𝑛). It follows from the equivalent definitions of Theorems 2.7 and 2.9 that the latter is equivalent to sup_{𝑎∈𝔹𝑛}4log1−|𝑎|^2(𝑝+1)loglog𝑒⁴1−|𝑎|^2𝑞_𝔹^𝑛1−|𝑎|^2𝑛|1−⟨𝑧,𝑎⟩|^2𝑛1+(2/s)𝑑𝜇(𝑧)<∞. (3.22) The proof is complete.

The methods used in this text are quite specific to the case considered here, that is, the embedding 𝐼_𝜇∶ℋ_𝑠(𝔹^𝑛)→𝐿^𝑝(𝔹^𝑛). 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 𝔹^𝑛 such that the embedding map 𝐼_𝜇∶ℋ^Φ1(𝔹^𝑛)→ℋ^Φ2(𝔹^𝑛) (Φ₁≠Φ₂ if Φ₁ and Φ₂ are convex growth functions) is bounded, is still open.

Acknowledgement

The author acknowledges support from the “Irish Research Council for Science, Engineering and Technology”.