Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2010, Article ID 603968, 7 pages
doi:10.1155/2010/603968
Research Article
Carleson Measure in Bergman-Orlicz
Space of Polydisc
An-Jian Xu1, 2 and Zou Yang3
1
Department of Mathematics, Zhejiang University, Hangzhou 310027, China
Institute of Applied Mathematics, Chongqing University of Post and Telecommunication,
Chongqing 400065, China
3
Department of Mathematics, Chongqing Education College, Chongqing 400067, China
2
Correspondence should be addressed to An-Jian Xu, anjian.xu@gmail.com
Received 30 June 2010; Revised 26 August 2010; Accepted 2 September 2010
Academic Editor: Dumitru Baleanu
Copyright q 2010 A.-J. Xu and Z. Yang. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution, and
reproduction in any medium, provided the original work is properly cited.
Let μ 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
ϕ
in order that the identity map J : La Dn , σn → Lϕ Dn , μ is bounded.
1. Introduction
We denote by Dn 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 SR to denote the corona
associated to these sets. In particular, if I is an interval on T of length δ ∈ 0, 1 centered at
eiθ0 δ/2 ,
SI {z ∈ D | 1 − δ < r < 1, θ0 < θ < θ0 δ}.
1.1
Then, if R I1 × I2 × · · · × In ⊂ Tn , with Ij intervals having length δj and having centers
eiθj δj /2 , SR is given by SR SI1 × SI2 × · · · × SIn , and let
0
0
αj 1 − δj eiθj δj /2 ,
1 ≤ j ≤ n.
1.2
If V is any open set in Tn , we define SV ∪γ SRγ where {Rγ } runs through all rectangles
in V .
2
Abstract and Applied Analysis
An Orlicz function is a real-valued, nondecreasing, convex function ϕ : 0, ∞ →
0, ∞ such that ϕ0 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
ϕx
∞.
x→∞ x
lim
1.3
The Orlicz space Lϕ μ is the space of all equivalence classes of measurable functions
f : Ω → C for which there is a constant C > 0 such that
fw
dμw < ∞,
ϕ
C
Ω
1.4
and then fμ 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
Mμ f :
Ω
ϕ f dμ < ∞.
1.5
ϕ
The Bergman-Orlicz space La 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 H p norm dominates the Lp 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 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 measure μ on Dn is called a ϕ Carleson measure if there is a constant
C such that
μSI ≤
1
n
2
−1
ϕ Cϕ
j1 1/δj
for every rectangle I ⊂ Tn .
In this paper, we proveTheorem 2.4.
1.6
Abstract and Applied Analysis
3
2. Main Results and Proofs
Lemma 2.1. For α α1 , . . . , αn ∈ Dn , let uα z1 , . . . , zn ϕ
uα z1 , . . . , zn ∈ La Dn , and
uα zσn ≤
ϕ−1
1
n
j1
1/δj2
n
j1 1
− |αj |2 2 /1 − αj zj 4 . Then
.
2.1
Proof. It is easy to see that uα z∞ nj1 1 |αj |/1 − |αj |2 nj1 2 − δj /δj 2 . Since
ϕ0 0, the convexity of ϕ implies ϕax ≤ aϕx for 0 ≤ a ≤ 1. Hence, for every C > 0, we
have
Dn
⎛
ϕ⎝
n j1 δj /2
⎞
2
2 n
− δj |uα z|
δj
1
⎠dσn ≤
dσn
|uα z|ϕ
C
2
−
δ
C
n
j
D
j1
n
j1
n
2
j1 δj /2 − δj uα1 z1 ϕ1/C
2
δj /δj 1/uα z1 , that is,
but
uα zσn ≤
ϕ−1
≤
n j1 2
δj
2 − δj
2
2.2
1
,
uα1 z1 ϕ
C
1 if and only if C
≥
1
.
2
− δj /δj 1/uα z1 1/ϕ−1 n
j1 2
−
2.3
Moreover,
uα z1 D
1 − |α1 |2
2
|1 − α1 z1 |4
n dσ1 z1 · · ·
2 2
1 − αj D
j1
n 1 − |αn |2
2 D
j1
1 − |αn |2
D
2
dσ1 zn |1 − αn zn |4
1
dσ1 zj
1 − αj zj 4
1
1
2 1 − αj zj
1 − αj zj
2 dσ1 zj
2.4
2
n
1 − |αn |2
2 2 1.
j1 1 − αj So, we have
uα zσn ≤
ϕ−1
1
n j1 2
− δj /δj
2 ≤
ϕ−1
1
n
j1
1/δj2
.
2.5
4
Abstract and Applied Analysis
For m m1 , . . . , mn ∈ Zn ,k k1 , . . . , kn with 1 ≤ kj ≤ 2mj 4 , 1 ≤ j ≤ n, let
Tmk r1 eiθ1 , . . . , rn eiθn | 1 − 2−mj ≤ rj < 1 − 2−mj −1 ,
2kj π
2mj 4
2kj 1 π
≤ θj <
2mj 4
2.6
,1 ≤ j ≤ n ,
mk
let zmk zmk
1 , . . . , zn , where
mj 4
zmk
1 − 2−mj e2kj 1/2πi/2 ,
j
1 ≤ j ≤ n,
2.7
let
Umk 7
−mj
2
≤
,
1
≤
j
≤
n
.
z1 , . . . , zn | z − zmk
j
8
2.8
Lemma 2.2. For fixed m0 m01 , . . . , m0n and the corresponding k0 k10 , . . . , kn0 , Tm0 k0 intersect
Umk for at most N 5.57n choices of the pair m, k.
Proof. See 5.
ϕ
Lemma 2.3. If f ∈ La Dn , then
n
ϕ fz1 , . . . , zn ≤ C1
j1
ϕ f dσn
Umk
2.9
for 6/82−mj ≤ ρj ≤ 7/82−mj and any z ∈ Tmk .
Proof. It is clear that ϕ|fz1 , . . . , zn | is an n-subharmonic function in Dn . Repeated
application of Harnack’s inequality yields
ϕ fz1 , . . . , zn ≤
1
2πn
≤ C2
2π
2π zj − zmk
j iθ1
mk
iθn ···
ϕ f zmk
ρ
e
,
.
.
.
,
z
ρ
e
dθ1 · · · dθn
1
n
n
1
0
0
ρj − zj − zmk
j n ρj
j1
1
2πn
2π
0
···
2π iθ1
mk
iθn ϕ f zmk
ρ
e
,
.
.
.
,
z
ρ
e
dθ1 · · · dθn .
1
n
n
1
0
2.10
Abstract and Applied Analysis
5
Hence, for z ∈ Tmk ,
⎛
⎞
7/82−m1
7/82−mn
n
ϕ fz1 , . . . , zn C2 ⎝ 4mj ⎠
···
ϕ fz ρ1 · · · ρn dρ1 · · · dρn
6/82−m1
j1
6/82−mn
2.11
⎛
⎞
n
≤ C2 C3 ⎝ 4mj ⎠
ϕ f dσn .
Umk
j1
Theorem 2.4 Main theorem. Let μ be a finite, positive measure on Dn , and suppose that ϕ is an
Orlicz function. Then, the identity map
ϕ
J : La Dn , σn −→ Lϕ Dn , 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
Jg ≤ Cg ,
σn
μ
2.13
2 2
n
1 − αj ϕ
n
uα z 4 ∈ La D .
α
z
1
−
j1
j j
2.14
ϕ
for all g ∈ La Dn . By Lemma 2.1,
However, for zj ∈ SIj , we have
1 − αj z ≤ 1 − αj eiθ0 δj /2 αj eiθ0 δj /2 − αj z
iθ δ /2 z z
0
j
≤ δj 1 − δ j
− z
−
e
|z| |z|
≤ δj 1 − δj δj 1 − |z|
≤ δj 2δj 1 − δj ≤ 3δj ,
2.15
so,
2 2
2
n
n
1 − αj 1 δj
1
1
≥ n.
|uα z1 , . . . , zn | 4 ≥ n
2
3
3
δj
j1 1 − αj zj
j1
2.16
6
Abstract and Applied Analysis
Therefore,
⎛
1≥
Dn
⎜
ϕ⎝
⎜
ϕ⎝
≥
n
j1
ϕ−1
n
j1
⎜
ϕ⎝
ϕ−1
1/δj2
|uα z|
1/δj2
n
j1
1/δj2
⎞
⎟
⎠dμ
⎞
⎟
⎠dμ
3n C
SI
⎛
C
⎛
ϕ−1
2.17
⎞
⎟
⎠μSI,
3n C
that is,
μSI ≤
1
,
ϕ−1 n
2
ϕ C
j1 1/δj
2.18
with C 1/3n C.
ϕ
Conversely, suppose that fz ∈ La Dn , we have
fz1 , . . . , cn dμ
ϕ
C1 NμDn f σn
Dn
mm1 ,...,mn ,mj ≥0
≤
m
k
mj 4
kk1 ,...,kn ,1≤kj ≤2
⎧
⎨
n μTmk C1
⎩ j1
fz1 , . . . , cn dμ
ϕ
C1 NμDn f σn
Tmk
⎫
⎬
fz1 , . . . , cn ϕ
dσ
n
⎭
C1 NμDn f σn
Umk
≤ C1 μD fz1 , . . . , cn dσn
ϕ
C1 NμDn f σn
Umk
m k
fz1 , . . . , cn n
dσn
ϕ
C1 μD C1 NμDn f T 0 0 ∩U
n
m,k m0 ,k0
m0 ,k0 m,k
≤ C1 μDn N
m0 ,k0
C1 NμDn and the proof is complete.
Tm0 k0 ∩Umk
Tm0 k0
σn
mk
C1 μDn m k
fz1 , . . . , cn dσn
ϕ
C1 NμDn f σn
fz1 , . . . , cn dσn
ϕ
C1 NμDn f σn
fz1 , . . . , cn dσn ≤ 1,
ϕ
C1 NμDn f σn
Dn
2.19
Abstract and Applied Analysis
7
Corollary 2.5. Let μ 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
uα zσn ≤
ϕ−1
C
n
j1
1/δj2
,
2.20
for every rectangle I ⊂ Dn .
Proof. As a fact, for any measure μ and Orlicz function ϕ, we have
1≥
Dn
ϕ
|uα z|
uα zσn
dμ ≥ ϕ
1
3n uα zσn
μSI
2.21
by the proof of the Main theorem. So,
μSI ≤
1
,
n
ϕ 1/3 uα zσn
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.
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