# Carleson Measure in Bergman-Orlicz Space of Polydisc

Authors: An-Jian Xu; Zou Yang
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

Letμ be a finite, positive measure on 𝔻n, the polydisc in ℂn, and let σn be 2n-dimensional Lebesgue volume measure on 𝔻n. For an Orlicz function φ, a necessary and sufficient condition on μ is given in order that the identity map J:Laφ(𝔻n,σn)→Lφ(𝔻n,μ) is bounded.


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## Body

## 1. Introduction

We denote by𝔻n the unit polydisc in ℂn and by 𝕋n the distinguished boundary of 𝔻n. We will use σn to denote the 2n-dimensional Lebesgue volume measure on 𝔻n, normalized so that σn(𝔻n)=1. We use R to describe rectangles on 𝕋n, 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 ei(θ0+δ/2), (1.1)S(I)={z∈𝔻∣1-δ<r<1,θ0<θ<θ0+δ}.
Then, if R=I1×I2×⋯×In⊂𝕋n, with Ij intervals having length δj and having centers ei(θj0+δj/2), S(R) is given by S(R)=S(I1)×S(I2)×⋯×S(In), and let (1.2)αj=(1-δj)ei(θj0+δj/2),1≤j≤n.
If V is any open set in 𝕋n, we define S(V)=∪γS(Rγ) where {Rγ} runs through all rectangles in V.An Orlicz function is a real-valued, nondecreasing, convex functionφ:[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 (1.3)limx→∞φ(x)x=∞.The Orlicz spaceLφ(μ) is the space of all (equivalence classes of) measurable functions f:Ω→ℂ for which there is a constant C>0 such that (1.4)∫Ωφ(|f(w)|C)dμ(w)<+∞,
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 Lφ(μ) is a Banach space under the Luxemburg norm ∥·∥μ. For f∈Lφ, let (1.5)Mμ(f)∶=∫Ωφ(|f|)dμ<+∞.The Bergman-Orlicz spaceLaφ(𝔻n,σn) consists of all analytic functions in Lφ(𝔻n,σn), which is a closed subspace of Lφ(𝔻n,σn), 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 Hp norm dominates the Lp(μ) norm of elements of Hp. 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 Lp(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 ℂn 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 measureμ on 𝔻n is called a φ Carleson measure if there is a constant C' such that (1.6)μ(S(I))≤1φ(C′φ-1(∏j=1n(1/δj2)))′
for every rectangle I⊂𝕋n.In this paper, we proveTheorem2.4.


## 2. Main Results and Proofs

Lemma 2.1.
Forα=(α1,…,αn)∈𝔻n, let uα(z1,…,zn)=∏j=1n(1-|αj|2)2/(1-α¯jzj)4. Then uα(z1,…,zn)∈Laφ(𝔻n), and
(2.1)∥uα(z)∥σn≤1φ-1(∏j=1n(1/δj2)).Proof.
It is easy to see that∥uα(z)∥∞=∏j=1n((1+|αj|)/(1-|αj|))2=∏j=1n((2-δj)/δj)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
(2.2)∫𝔻nφ(∏j=1n(δj/(2-δj))2|uα(z)|C)dσn≤∏j=1n(δj2-δj)2∫𝔻n|uα(z)|φ(1C)dσn=∏j=1n(δj2-δj)2∥uα1(z)∥1φ(1C),
but ∏j=1n(δj/(2-δj))2∥uα1(z)∥1φ(1/C)≤1 if and only if C≥1/φ-1(∏j=1n(2-(δj/δj))2(1/∥uα(z)∥1)), that is,
(2.3)∥uα(z)∥σn≤1φ-1(∏j=1n(2-δj/δj)2(1/∥uα(z)∥1)).
Moreover,
(2.4)∥uα(z)∥1=∫𝔻(1-|α1|2)2|1-α¯1z1|4dσ1(z1)⋯∫𝔻(1-|αn|2)2|1-α¯nzn|4dσ1(zn)=∏j=1n(1-|αj|2)2∫𝔻1|1-α¯jzj|4dσ1(zj)=∏j=1n(1-|αn|2)2∫𝔻1(1-α¯jzj)21(1-αjzj¯)2dσ1(zj)=∏j=1n(1-|αn|2)2(1-|αj|2)2=1.
So, we have
(2.5)∥uα(z)∥σn≤1φ-1(∏j=1n((2-δj)/δj)2)≤1φ-1(∏j=1n(1/δj2)).
Form=(m1,…,mn)∈ℤ+n,k=(k1,…,kn) with 1≤kj≤2mj+4,(1≤j≤n), let
(2.6)Tmk={(2kj+1)π2mj+4(r1eiθ1,…,rneiθn)∣1-2-mj≤rj<1-2-mj-1,2kjπ2mj+4≤θj<(2kj+1)π2mj+4,1≤j≤n},
let zmk=(z1mk,…,znmk), where
(2.7)zjmk=(1-2-mj)e2(kj+1/2)πi/2mj+4,1≤j≤n,
let
(2.8)Umk={(z1,…,zn)∣|z-zjmk|≤782-mj,1≤j≤n}.Lemma 2.2.
For fixedm0=(m10,…,mn0) and the corresponding k0=(k10,…,kn0), Tm0k0 intersect Umk for at most N=(5.57)n choices of the pair (m,k).Proof.
See [5].Lemma 2.3.
Iff∈Laφ(𝔻n), then
(2.9)φ(|f(z1,…,zn)|)≤C1∏j=1n∫Umkφ(|f|)dσn
for (6/8)2-mj≤ρj≤(7/8)2-mjand any z∈Tmk.Proof.
It is clear thatφ(|f(z1,…,zn)|) is an n-subharmonic function in 𝔻n. Repeated application of Harnack's inequality yields
(2.10)φ(|f(z1,…,zn)|)≤1(2π)n∏j=1nρj+|zj-zjmk|ρj-|zj-zjmk|∫02π⋯∫02πφ(|f(z1mk+ρ1eiθ1,…,znmk+ρneiθn)|)dθ1⋯dθn≤C21(2π)n∫02π⋯∫02πφ(|f(z1mk+ρ1eiθ1,…,znmk+ρneiθn)|)dθ1⋯dθn.
Hence, for z∈Tmk,
(2.11)φ(|f(z1,…,zn)|)=C2(∏j=1n4mj)∫(6/8)2-m1(7/8)2-m1⋯∫(6/8)2-mn(7/8)2-mnφ(|f(z)|)ρ1⋯ρndρ1⋯dρn≤C2C3(∏j=1n4mj)∫Umkφ(|f|)dσn.Theorem 2.4 (Main theorem).
Letμ be a finite, positive measure on 𝔻n, and suppose that φ is an Orlicz function. Then, the identity map
(2.12)J:Laφ(𝔻n,σn)→Lφ(𝔻n,μ)
is bounded if and only if μ is a φ Carleson measure.Proof.
Suppose that there exists a constantC such that
(2.13)∥J(g)∥σn≤C∥g∥μ,
for all g∈Laφ(𝔻n). By Lemma 2.1,
(2.14)uα(z)=∏j=1n(1-|αj|2)2(1-αj¯zj)4∈Laφ(𝔻n).
However, for zj∈S(Ij), we have
(2.15)|1-α¯jz|≤|1-α¯jei(θ0+δj/2)|+|α¯jei(θ0+δj/2)-α¯jz|≤δj+(1-|δj|)(|ei(θ0+δj/2)-z|z||+|z|z|-z|)≤δj+(1-δj)(δj+(1-|z|))≤δj+2δj(1-δj)≤3δj,
so,
(2.16)|uα(z1,…,zn)|=∏j=1n(1-|αj|2)2|1-αj¯zj|4≥13n∏j=1n(1+δj)2δj2≥13n.
Therefore,
(2.17)1≥∫𝔻nφ(φ-1(∏j=1n(1/δj2))|uα(z)|C)dμ≥∫S(I)φ(φ-1(∏j=1n(1/δj2))3nC)dμ=φ(φ-1(∏j=1n(1/δj2))3nC)μ(S(I)),
that is,
(2.18)μ(S(I))≤1φ(C'φ-1(∏j=1n(1/δj2))),
with C′=1/3nC.
Conversely, suppose thatf(z)∈Laφ(𝔻n), we have
(2.19)∫𝔻nφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dμ=∑m=(m1,…,mn),mj≥0‍∑k=(k1,…,kn),1≤kj≤2mj+4∫Tmkφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dμ≤∑m∑kμ(Tmk){C1∏j=1n∫Umkφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}≤C1μ(𝔻n)∑m∑k{∫Umkφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}=C1μ(𝔻n){∑m,k∑m0,k0∫Tm0k0∩Umkφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}=C1μ(𝔻n){∑m0,k0∑m,k∫Tm0k0∩Umkφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}≤C1μ(𝔻n){N∑m0,k0∫Tm0k0φ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}=C1Nμ(𝔻n){∫𝔻nφ(|f(z1,…,cn)|C1Nμ(𝔻n)∥f∥σn)dσn}≤1,
and the proof is complete.Corollary 2.5.
Letμ be a finite, positive measure on 𝔻n, 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
(2.20)∥uα(z)∥σn≤Cφ-1(∏j=1n(1/δj2)),
for every rectangle I⊂𝔻n.Proof.
As a fact, for any measureμ and Orlicz function φ, we have
(2.21)1≥∫𝔻nφ(|uα(z)|∥uα(z)∥σn)dμ≥φ(13n∥uα(z)∥σn)μ(S(I))
by the proof of the Main theorem. So,
(2.22)μ(S(I))≤1φ(1/3n∥uα(z)∥σn),
and the corollary follows.



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*Source: 603968-2010-09-27.xml*