Carleson Measure in Bergman-Orlicz Space of Polydisc

Abstract

Let 𝜇 be a finite, positive measure on 𝔻^𝑛, the polydisc in ℂ^𝑛, and let 𝜎_𝑛 be 2n-dimensional Lebesgue volume measure on 𝔻^𝑛. For an Orlicz function 𝜑, a necessary and sufficient condition on 𝜇 is given in order that the identity map 𝐽∶𝐿^𝜑_𝑎(𝔻^𝑛,𝜎_𝑛)→𝐿^𝜑(𝔻^𝑛,𝜇) is bounded.

1. Introduction

We denote by 𝔻^𝑛 the unit polydisc in ℂ^𝑛 and by 𝕋^𝑛 the distinguished boundary of 𝔻^𝑛. We will use 𝜎_𝑛 to denote the 2𝑛-dimensional Lebesgue volume measure on 𝔻^𝑛, normalized so that 𝜎_𝑛(𝔻^𝑛)=1. We use 𝑅 to describe rectangles on 𝕋^𝑛, and we use 𝑆(𝑅) to denote the corona associated to these sets. In particular, if I is an interval on 𝕋 of length 𝛿∈(0,1) centered at 𝑒^{𝑖(𝜃0+𝛿/2)}, 𝑆(𝐼)={𝑧∈𝔻∣1−𝛿<𝑟<1,𝜃₀<𝜃<𝜃₀+𝛿}. (1.1) Then, if 𝑅=𝐼₁×𝐼₂×⋯×𝐼_𝑛⊂𝕋^𝑛, with 𝐼_𝑗 intervals having length 𝛿_𝑗 and having centers 𝑒^{𝑖(𝜃0𝑗+𝛿𝑗/2)}, 𝑆(𝑅) is given by 𝑆(𝑅)=𝑆(𝐼₁)×𝑆(𝐼₂)×⋯×𝑆(𝐼_𝑛), and let 𝛼_𝑗=(1−𝛿_𝑗)𝑒^{𝑖(𝜃0𝑗+𝛿𝑗/2)},1≤𝑗≤𝑛. (1.2) If 𝑉 is any open set in 𝕋^𝑛, we define 𝑆(𝑉)=∪_𝛾𝑆(𝑅_𝛾) where {𝑅_𝛾} runs through all rectangles in 𝑉.

An Orlicz function is a real-valued, nondecreasing, convex function 𝜑∶[0,+∞)→[0,+∞) such that 𝜑(0)=0 and 𝜑(∞)=∞. To avoid pathologies, we will assume that we work with an Orlicz function 𝜑 having the following additional properties: 𝜑 is continuous and strictly convex (hence increasing), such that lim_𝑥→∞𝜑(𝑥)𝑥=∞. (1.3)

The Orlicz space 𝐿^𝜑(𝜇) is the space of all (equivalence classes of) measurable functions 𝑓∶Ω→ℂ for which there is a constant 𝐶>0 such that _Ω𝜑(|𝑓(𝑤)|𝐶)𝑑𝜇(𝑤)<+∞, (1.4) and then ‖𝑓‖_𝜇 (the Luxemburg norm) is the infimum of all possible constant 𝐶 such that this integral is ≤1. It is well known that 𝐿^𝜑(𝜇) is a Banach space under the Luxemburg norm ‖⋅‖_𝜇. For 𝑓∈𝐿^𝜑, let 𝑀_𝜇(𝑓)∶=_Ω𝜑(|𝑓|)𝑑𝜇<+∞. (1.5)

The Bergman-Orlicz space 𝐿^𝜑_𝑎(𝔻^𝑛,𝜎_𝑛) consists of all analytic functions in 𝐿^𝜑(𝔻^𝑛,𝜎_𝑛), which is a closed subspace of 𝐿^𝜑(𝔻^𝑛,𝜎_𝑛), so it is an analytic Banach space also.

A theorem of Carleson [1, 2] characterizes those positive measure 𝜇 on 𝔻 for which the Hardy space 𝐻^𝑝 norm dominates the 𝐿^𝑝(𝜇) norm of elements of 𝐻^𝑝. Since then, there is a long history of the development and application of Carleson measures, see [3]. This rich area of research contains a large body of literature on characterizations of different classes of operators in different spaces and their applications. Chang [4] has characterized the bounded measures on 𝐿^𝑝(𝕋²) using a two-line proof referring to a result of Stein. Characterization of the bounded identity operators on Hardy spaces is an immediate consequence of Chang's proof using standard arguments. Hastings [5] has given a similar result for unweighted Bergman spaces. MacCluer [6] has obtained a Carleson measure characterization of the identity operators on Hardy spaces of the unit ball in ℂ^𝑛 using the well-known results of Hormander. Lefèvre et al. [7] have introduced an adapted version of Carleson measure in Hardy-Orlicz spaces. Xiao [8], Ortiz, and Fernandez [9] have got a characterization of the Carleson measure in Bergman-Orlicz spaces of the unit disc.

A finite, positive measure 𝜇 on 𝔻^𝑛 is called a 𝜑 Carleson measure if there is a constant 𝐶^′ such that 1𝜇(𝑆(𝐼))≤𝜑(𝐶^′𝜑⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗)))^′ (1.6) for every rectangle 𝐼⊂𝕋^𝑛.

In this paper, we proveTheorem 2.4.

2. Main Results and Proofs

Lemma 2.1. For 𝛼=(𝛼₁,…,𝛼_𝑛)∈𝔻^𝑛, let 𝑢_𝛼(𝑧₁,…,𝑧_𝑛)=∏^𝑛_𝑗=1(1−|𝛼_𝑗|²)²/(1−𝛼_𝑗𝑧_𝑗)⁴. Then 𝑢_𝛼(𝑧₁,…,𝑧_𝑛)∈𝐿^𝜑_𝑎(𝔻^𝑛), and ‖𝑢_𝛼(𝑧)‖_𝜎𝑛≤1𝜑⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗.)) (2.1)

Proof. It is easy to see that ‖𝑢_𝛼(𝑧)‖_∞=∏^𝑛_𝑗=1((1+|𝛼_𝑗|)/(1−|𝛼_𝑗|))²=∏^𝑛_𝑗=1((2−𝛿_𝑗)/𝛿_𝑗)². Since 𝜑(0)=0, the convexity of 𝜑 implies 𝜑(𝑎𝑥)≤𝑎𝜑(𝑥) for 0≤𝑎≤1. Hence, for every 𝐶>0, we have _𝔻𝑛∏𝜑(^𝑛𝑗=1(𝛿_𝑗/(2−𝛿_𝑗))²|𝑢_𝛼(𝑧)|𝐶)𝑑𝜎_𝑛≤^𝑛_𝑗=1(𝛿_𝑗2−𝛿_𝑗)²_𝔻𝑛|𝑢_𝛼1(𝑧)|𝜑(𝐶)𝑑𝜎_𝑛=^𝑛_𝑗=1(𝛿_𝑗2−𝛿_𝑗)²‖𝑢_𝛼1(𝑧)‖₁1𝜑(𝐶), (2.2) but ∏^𝑛_𝑗=1(𝛿_𝑗/(2−𝛿_𝑗))²‖𝑢_𝛼1(𝑧)‖₁𝜑(1/𝐶)≤1 if and only if 𝐶≥1/𝜑⁻¹(∏^𝑛_𝑗=1(2−(𝛿_𝑗/𝛿_𝑗))²(1/‖𝑢_𝛼(𝑧)‖₁)), that is, ‖𝑢_𝛼(z)‖_𝜎𝑛≤1𝜑⁻¹(∏^𝑛_𝑗=1(2−𝛿_𝑗/𝛿_𝑗)²(1/‖𝑢_𝛼(𝑧)‖₁.)) (2.3) Moreover, ‖𝑢_𝛼(𝑧)‖₁=_𝔻(1−|𝛼₁|²)²|1−𝛼₁𝑧₁|⁴𝑑𝜎₁(𝑧₁)⋯_𝔻(1−|𝛼_𝑛|²)²|1−𝛼_𝑛𝑧_𝑛|⁴𝑑𝜎₁(𝑧_𝑛)=^𝑛_𝑗=1(1−|𝛼_𝑗|²)²_𝔻1|1−𝛼_𝑗𝑧_𝑗|⁴𝑑𝜎₁(𝑧_𝑗)=^𝑛_𝑗=1(1−|𝛼_𝑛|²)²_𝔻1(1−𝛼_𝑗𝑧_𝑗)²1(1−𝛼_𝑗𝑧_𝑗)²𝑑𝜎₁(𝑧_𝑗)=^𝑛_𝑗=1(1−|𝛼_𝑛|²)²(1−|𝛼_𝑗|²)²=1. (2.4) So, we have ‖𝑢_𝛼(𝑧)‖_𝜎𝑛≤1𝜑⁻¹(∏^𝑛_𝑗=1((2−𝛿_𝑗)/𝛿_𝑗)²)≤1𝜑⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗.)) (2.5)
For 𝑚=(𝑚₁,…,𝑚_𝑛)∈ℤ^𝑛₊,𝑘=(𝑘₁,…,𝑘_𝑛) with 1≤𝑘_𝑗≤2^𝑚𝑗+4,(1≤𝑗≤𝑛), let 𝑇_𝑚𝑘={(𝑟₁𝑒^𝑖𝜃1,…,𝑟_𝑛𝑒^𝑖𝜃𝑛)∣1−2^−𝑚𝑗≤𝑟_𝑗<1−2^{−𝑚𝑗−1},2𝑘_𝑗𝜋2^𝑚𝑗+4≤𝜃_𝑗<(2𝑘_𝑗+1)𝜋2^𝑚𝑗+4,1≤𝑗≤𝑛}, (2.6) let 𝑧^𝑚𝑘=(𝑧₁^𝑚𝑘,…,𝑧_𝑛^𝑚𝑘), where 𝑧_𝑗^𝑚𝑘=(1−2^−𝑚𝑗)𝑒^{2(𝑘𝑗+1/2)𝜋𝑖/2𝑚𝑗+4},1≤𝑗≤𝑛, (2.7) let 𝑈_𝑚𝑘={(𝑧₁,…,𝑧_𝑛)∣|𝑧−𝑧_𝑗^𝑚𝑘7|≤82^−𝑚𝑗,1≤𝑗≤𝑛}. (2.8)

Lemma 2.2. For fixed 𝑚⁰=(𝑚⁰₁,…,𝑚⁰_𝑛) and the corresponding 𝑘⁰=(𝑘⁰₁,…,𝑘⁰_𝑛), 𝑇_𝑚0𝑘⁰ intersect 𝑈_𝑚𝑘 for at most 𝑁=(5.57)^𝑛 choices of the pair (𝑚,𝑘).

Proof. See [5].

Lemma 2.3. If 𝑓∈𝐿^𝜑_𝑎(𝔻^𝑛), then 𝜑(|𝑓(𝑧₁,…,𝑧_𝑛)|)≤𝐶₁^𝑛_{𝑗=1𝑈𝑚𝑘}𝜑(|𝑓|)𝑑𝜎_𝑛 (2.9) for (6/8)2^−𝑚𝑗≤𝜌_𝑗≤(7/8)2^−𝑚𝑗and any 𝑧∈𝑇_𝑚𝑘.

Proof. It is clear that 𝜑(|𝑓(𝑧₁,…,𝑧_𝑛)|) is an n-subharmonic function in 𝔻^𝑛. Repeated application of Harnack's inequality yields 𝜑(|𝑓(𝑧₁,…,𝑧_𝑛≤1)|)(2𝜋)^𝑛𝑛_𝑗=1𝜌_𝑗+|𝑧_𝑗−𝑧_𝑗^𝑚𝑘|𝜌_𝑗−|𝑧_𝑗−𝑧_𝑗^𝑚𝑘|₀^2𝜋⋯₀^2𝜋𝜑(|𝑓(𝑧₁^𝑚𝑘+𝜌₁𝑒^𝑖𝜃1,…,𝑧_𝑛^𝑚𝑘+𝜌_𝑛𝑒^𝑖𝜃𝑛)|)𝑑𝜃₁⋯𝑑𝜃_𝑛≤𝐶₂1(2𝜋)^𝑛₀^2𝜋⋯₀^2𝜋𝜑(|𝑓(𝑧₁^𝑚𝑘+𝜌₁𝑒^𝑖𝜃1,…,𝑧_𝑛^𝑚𝑘+𝜌_𝑛𝑒^𝑖𝜃𝑛)|)𝑑𝜃₁⋯𝑑𝜃_𝑛. (2.10) Hence, for 𝑧∈𝑇_𝑚𝑘, 𝜑(|𝑓(𝑧₁,…,𝑧_𝑛)|)=𝐶₂(^𝑛_𝑗=14^𝑚𝑗)^{(7/8)21−𝑚}_{(6/8)21−𝑚}⋯^{(7/8)2𝑛−𝑚}_{(6/8)2𝑛−𝑚}𝜑(|𝑓(𝑧)|)𝜌₁⋯𝜌_𝑛𝑑𝜌₁⋯𝑑𝜌_𝑛≤𝐶₂𝐶₃(^𝑛_𝑗=14^𝑚𝑗)_𝑈𝑚𝑘𝜑(|𝑓|)𝑑𝜎_𝑛. (2.11)

Theorem 2.4 (Main theorem). Let 𝜇 be a finite, positive measure on 𝔻^𝑛, and suppose that 𝜑 is an Orlicz function. Then, the identity map 𝐽∶𝐿^𝜑_𝑎(𝔻^𝑛,𝜎_𝑛)→𝐿^𝜑(𝔻^𝑛,𝜇) (2.12) is bounded if and only if 𝜇 is a 𝜑 Carleson measure.

Proof. Suppose that there exists a constant 𝐶 such that ‖𝐽(𝑔)‖_𝜎𝑛≤𝐶‖𝑔‖_𝜇, (2.13) for all 𝑔∈𝐿^𝜑_𝑎(𝔻^𝑛). By Lemma 2.1, 𝑢_𝛼(𝑧)=^𝑛_𝑗=1(1−|𝛼_𝑗|²)²(1−𝛼_𝑗𝑧_𝑗)⁴∈𝐿^𝜑_𝑎(𝔻^𝑛). (2.14) However, for 𝑧_𝑗∈𝑆(𝐼_𝑗), we have |1−𝛼_𝑗𝑧|≤|1−𝛼_𝑗𝑒^{𝑖(𝜃0+𝛿𝑗/2)}|+|𝛼_𝑗𝑒^{𝑖(𝜃0+𝛿𝑗/2)}−𝛼_𝑗𝑧|≤𝛿_𝑗+(1−|𝛿_𝑗|)(|𝑒^{𝑖(𝜃0+𝛿𝑗/2)}−𝑧𝑧|𝑧||+||𝑧|−𝑧|)≤𝛿_𝑗+(1−𝛿_𝑗)(𝛿_𝑗+(1−|𝑧|))≤𝛿_𝑗+2𝛿_𝑗(1−𝛿_𝑗)≤3𝛿_𝑗, (2.15) so, |𝑢_𝛼(𝑧₁,…,𝑧_𝑛)|=^𝑛_𝑗=1(1−|𝛼_𝑗|²)²|1−𝛼_𝑗𝑧_𝑗|⁴≥13^𝑛𝑛_𝑗=1(1+𝛿_𝑗)²𝛿²_𝑗≥13^𝑛. (2.16) Therefore, 1≥_𝔻𝑛𝜑𝜑(⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗))|𝑢_𝛼(𝑧)|𝐶≥)𝑑𝜇_𝑆(𝐼)𝜑𝜑(⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗))3^𝑛𝐶𝜑)𝑑𝜇=𝜑(⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗))3^𝑛𝐶)𝜇(𝑆(𝐼)), (2.17) that is, 1𝜇(𝑆(𝐼))≤𝜑(𝐶^′𝜑−1(∏^𝑛_𝑗=1(1/𝛿²_𝑗,))) (2.18) with 𝐶^′=1/3^𝑛𝐶.
Conversely, suppose that 𝑓(𝑧)∈𝐿^𝜑_𝑎(𝔻^𝑛), we have _𝔻𝑛𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜇=_{𝑚=(𝑚1,…,𝑚𝑛),𝑚𝑗≥0𝑘=(𝑘1,…,𝑘𝑛),1≤𝑘𝑗≤2𝑚𝑗+4𝑇𝑚𝑘}𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜇≤_𝑚𝑘𝜇(𝑇_𝑚𝑘){𝐶₁^𝑛_{𝑗=1𝑈𝑚𝑘}𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}≤𝐶₁𝜇(𝔻^𝑛)_𝑚𝑘{_𝑈𝑚𝑘𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}=𝐶₁𝜇(𝔻^𝑛){_{𝑚,𝑘𝑚0,𝑘0𝑇𝑚0𝑘0∩𝑈𝑚𝑘}𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}=𝐶₁𝜇(𝔻^𝑛){_{𝑚0,𝑘0𝑚,𝑘𝑇𝑚0𝑘0∩𝑈𝑚𝑘}𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}≤𝐶₁𝜇(𝔻^𝑛){𝑁_{𝑚0,𝑘0𝑇𝑚0𝑘0}𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁N𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}=𝐶₁𝑁𝜇(𝔻^𝑛){_𝔻𝑛𝜑(|𝑓(𝑧₁,…,𝑐_𝑛)|𝐶₁𝑁𝜇(𝔻^𝑛)‖𝑓‖_𝜎𝑛)𝑑𝜎_𝑛}≤1, (2.19) and the proof is complete.

Corollary 2.5. Let 𝜇 be a finite, positive measure on 𝔻^𝑛, and suppose that 𝜑 is an Orlicz function. Then 𝜇 is a 𝜑 Carleson measure if and only if there exists some 𝐶>1 such that ‖𝑢_𝛼(𝑧)‖_𝜎𝑛≤𝐶𝜑⁻¹(∏^𝑛_𝑗=1(1/𝛿²_𝑗,)) (2.20) for every rectangle 𝐼⊂𝔻^𝑛.

Proof. As a fact, for any measure 𝜇 and Orlicz function 𝜑, we have 1≥_𝔻𝑛𝜑(|𝑢_𝛼(𝑧)|‖𝑢_𝛼(𝑧)‖_𝜎𝑛1)𝑑𝜇≥𝜑(3^𝑛‖𝑢_𝛼(𝑧)‖_𝜎𝑛)𝜇(𝑆(𝐼)) (2.21) by the proof of the Main theorem. So, 1𝜇(𝑆(𝐼))≤𝜑(1/3^𝑛‖𝑢_𝛼(𝑧)‖_𝜎𝑛), (2.22) and the corollary follows.

Acknowledgment

This paper was supported by the NNSF of China (10671083) and the Youth Foundation of CQUPT (A2007-28).