Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2012, Article ID 583156, 23 pages
doi:10.1155/2012/583156
Research Article
Carleson Measure and Tent Spaces on the
Siegel Upper Half Space
Kai Zhao
College of Mathematics, Qingdao University, Qingdao 266071, China
Correspondence should be addressed to Kai Zhao, zkzc@yahoo.com.cn
Received 22 April 2012; Accepted 31 October 2012
Academic Editor: Natig Atakishiyev
Copyright q 2012 Kai Zhao. 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.
The Hausdorff capacity on the Heisenberg group is introduced. The Choquet integrals with respect
to the Hausdorff capacity on the Heisenberg group are defined. Then the fractional Carleson
measures on the Siegel upper half space are discussed. Some characterized results and the dual
of the fractional Carleson measures on the Siegel upper half space are studied. Therefore, the
tent spaces on the Siegel upper half space in terms of the Choquet integrals are introduced and
investigated. The atomic decomposition and the dual spaces of the tent spaces are obtained at the
last.
1. Introduction
It is well known that harmonic analysis plays an important role in partial differential
equations. The theory of function spaces constitutes an important part of harmonic analysis.
Heisenberg group, just as its name coming from the physicists Heisenberg, is very useful
in quantum mechanics. Therefore, to discuss new function or distribution spaces and some
characterizations of them is very significant in modern harmonic analysis and partial differential equations. Especially, the function spaces related to the Heisenberg group will be
used in partial differential equations and physics. It is precisely the reason in which we are
interested.
The Hausdorff capacity on Rn is introduced by Adams in 1. Some limiting form is the
classical Hausdorff measure. Adams also discussed some boundedness of Hardy-Littlewood
maximal functions related to it. The capacity and Choquet integrals, in some sense, are from
and applied to partial differential equations see 2, 3. As we know, the tent spaces have
been considered by many authors and play an important role in harmonic analysis on Rn cf.
4, 5. By using the Choquet integrals with respect to Hausdorff capacity on Rn , the new
tent spaces and their applications on the duality results for fractional Carleson measures,
2
Abstract and Applied Analysis
Q spaces, and Hardy-Hausdorff spaces were discussed see 6. The theory of Q spaces
can be found in 7–11. Inspired by the literature 6, in this paper, we will discuss the
fractional Carleson measures and the tent spaces on the Siegel upper half space. In order
to get the results in Section 3, some boundedness of Hardy-Littlewood maximal functions
q
on the Choquet integral space LΛ∞p Hn with Hausdorff capacity on the Heisenberg group
are discussed in Section 2. In Section 3, by the Choquet integrals with respect to Hausdorff
capacity on the Heisenberg group, we will introduce the fractional Carleson measures on
the Siegel upper half space. The characterizations of the fractional Carleson measures on the
Siegel upper half space are obtained. And the dual of the fractional Carleson measures on
the Siegel upper half space is also obtained. In the last section, the tent spaces on the Siegel
upper half space in terms of Choquet integrals with respect to the Hausdorff capacity on
the Heisenberg group are introduced. Then, the atomic decomposition and the dual spaces
of the tent spaces are obtained. The fractional Carleson measures and the tent spaces on the
Siegel upper half space will be used for Q spaces and the Hardy-Hausdorff spaces on the
Heisenberg group which will be discussed in another paper by us. On the other hand, they
may be used in partial differential equations and quantum mechanics.
As we know, Heisenberg group was discussed by many authors, such as 5, 12–15.
For convenience, let us recall some basic knowledge for the Heisenberg group.
Let Z, R, and C be the sets of all integers, real numbers, and complex numbers, and
n
n
Z , R , and Cn be n-dimensional Z, R, and C, respectively. The Siegel upper half space Un in
Cn1 is defined by
2 Un z z , zn1 ∈ Cn1 : Im zn1 > z ,
where z z1 , z2 , . . . , zn ∈ Cn , |z |2 n
k1
1.1
|zk |2 . The boundary of Un is
2 ∂Un z z , zn1 ∈ Cn1 : Im zn1 z .
1.2
The Heisenberg group on Cn × R, denoted by Hn , is a noncommutative nilpotent Lie
group with the underlying manifold Cn × R. The group law is given by
zw z , t w , s z w , t s 2 Im z w ,
for z z , t , w w , s ∈ Hn ,
1.3
where z w z1 w1 · · · zn wn is the Hermitean product on Cn .
It is easy to check that the inverse of the element z z , t is z−1 −z , −t, and
the unitary element is 0 0, 0. The Haar measure dz on Hn coincides with the Lebesgue
measure dz dt on Cn × R.
For each element ζ ζ , t of Hn , the following affine self-mapping of Un :
2 ζ , t : z , zn1 −→ z ζ , zn1 t 2iz ζ iζ 1.4
is an action of the group Hn on the space Un . Observe that the mapping 1.4 gives us a
realization of Hn as a group of affine holomorphic bijections of Un see 5.
Abstract and Applied Analysis
3
The dilations on Hn are defined by δz δz , t δz , δ2 t, δ > 0, and the rotation on
H is defined by σz σz , t σz , t with a unitary map σ of Cn . The conjugation of z is
z z , t z , −t. The norm function is given by
n
1/4
4
,
|z| z |t|2
z z , t ∈ Hn ,
1.5
which is homogeneous of degree 1 and satisfies |z−1 | |z| and |zw| ≤ C|z| |w| for some
absolute constant C. The distance function dz, w of point z and w in Hn is defined by
dz, w |w−1 z|.
For z z , t ∈ Hn or z x, y, t ∈ Hn for x x1 , x2 , , . . . , xn , y y1 , y2 , , . . . , yn ,
we define
|z|∞ max |x1 |, . . . , |xn |, y1 , . . . , yn , |t| ,
1.6
where z z1 , . . . , zn , and zk xk iyk , 1 ≤ k ≤ n.
To define the cube in Hn , let x x1 , x2 , . . . , xn , where a denotes the largest
integer less than or equal to a, and x x − x. We define the function Fx, y by
∞
2 j j n
j
j
n
−
2
,
2
x
2
y
mod
y
2
x
mod
F x, y 2Z
2Z
j
j1 4
1.7
and set
I0 x, y, t ∈ Hn : 0 ≤ xk ≤ 1, 0 ≤ yk ≤ 1, 1 ≤ k ≤ n, 0 ≤ t − F x, y ≤ 1 ,
1.8
Γ {m, n, l ∈ Hn : m, n ∈ Zn ; l ∈ Z}. Then
Hn γ −1 I0 ,
disjoint union ,
γ∈Γ
1.9
where I0 {x, y, t ∈ Hn : 0 ≤ xk < 1, 0 ≤ yk < 1, k 1, 2, . . . , n, 0 ≤ t − Fx, y < 1} see
14. Now a “cube” in fact, called “tile” more precisely with center w u, v, s and edge
sidelength l is defined by I lw−1 I0 . Let |I| with the Lebesgue measure be the volume of I
with length lI. It is easy to see that |I| lI2n2 . Obviously, the diameter of I denoted by
diamI is equal to cn lI, where cn is a constant depending only on n. The “dyadic cubes”
on Hn can be defined by I 2−j γ −1 I0 , γ ∈ Γ, j ∈ Z. We also call the “cube” and “dyadic cube”
as cube and dyadic cube, respectively.
A ball in Hn with center w and radius r is denoted as B Bw, r {z : |w−1 z| < r}.
The tent based on the set E ⊂ Hn is defined by
T E ξ, ρ ∈ Un : B ξ, ρ ⊂ E .
1.10
The Schwartz class of rapidly decreasing smooth function on Hn will be denoted by
SH . The dual of SHn is S Hn , the space of tempered distributions on Hn .
n
4
Abstract and Applied Analysis
2. Hausdorff Capacity on Heisenberg Group
In order to discuss the fractional Carleson measures on the Siegel upper half space, we
introduce the p-dimensional Hausdorff capacity and Choquet integral on the Heisenberg
group in the following.
Definition 2.1. For p ∈ 0, 1 and E ⊂ Hn , the p-dimensional Hausdorff capacity of E is defined
by
⎫
⎧
∞ ⎬
⎨ I p : E ⊂ Ij ,
Λ∞
p E inf
⎭
⎩ j j
j1
2.1
where the infimum ranges only over covers of E by dyadic cubes.
Remark 2.2. It is easy to check that Λ∞
p satisfies the following properties.
∞
i If Ej is nondecreasing, then Λ∞
p j Ej limj → ∞ Λp Ej .
∞
ii If Ej is nonincreasing, then Λ∞
p j Ej limj → ∞ Λp Ej .
iii Λ∞
p has the strong subadditivity condition.
∞
∞
E2 Λ∞
Λ∞
p E1
p E1 ∩ E2 ≤ Λp E1 Λp E2 .
2.2
For a nonnegative function f on Hn , the Choquet integral with respect to the Hausdorff
capacity on the Heisenberg group is defined by
Hn
fzdΛ∞
p
∞
0
Λ∞
p
z ∈ Hn : fz > λ dλ.
2.3
Since Λ∞
p is monotonous, the integrand on the right hand side in 2.3 is nonincreasing
and then is Lebesgue measurable on 0, ∞. It is easy to see that the equality 2.3 with
the Hausdorff measure is similar to the equality of the distribution function with Lebesgue
measure.
In order to characterize the fractional Carleson measure in the Siegel upper half space
in Section 3, we need to discuss the Hardy-Littlewood maximal operator on the Heisenberg
group.
Let f ∈ L1loc Hn . Mf, the dyadic Hardy-Littlewood maximal operator of f, is
defined by
1
M f z sup
Iz |I|
I
fξdξ,
2.4
where the supremum is taken over all dyadic cubes I containing z.
The boundedness of the dyadic Hardy-Littlewood maximal operators on the Choquet
q
integral spaces LΛ∞p Hn is as follows.
Abstract and Applied Analysis
5
Theorem 2.3. Suppose 0 < p ≤ 1. There exists a constant C depending only on p and n such that
Hn
Λ∞
p
q
M f dΛ∞
p ≤C
Hn
q ∞
f dΛp ,
C
ξ ∈ Hn : M f ξ > λ ≤ p
λ
Hn
q
∀f ∈ LΛ∞p Hn , q > p,
p ∞
f dΛp ,
p
∀f ∈ LΛ∞p Hn .
2.5
2.6
To prove Theorem 2.3, we need to prove the following lemmas first.
Lemma 2.4. For p ∈ 0, 1, let {Ij } be a sequence of dyadic cubes in Hn such that j |Ij |p < ∞.
Then there is a sequence of disjoint dyadic cubes {Jk } so that k Jk j Ij and k |Jk |p ≤ j |Ij |p .
Moreover, if E ⊂ j Ij , then the following tent inclusion T E ⊂ k T Jk∗ holds, where Jk∗ is the cube
with the same center as Jk and δn times the sidelength (δn is some constant depending only on n).
Proof. Since j |Ij |p < ∞, p ∈ 0, 1, we have that j |Ij | < ∞. Thus the union of Ij cannot form
arbitrarily large dyadic cubes. Thus, each Ij must be included in some maximal dyadic cube
Jk ∈ J, where J denotes the collection of all dyadic cubes J {Ij : Ij ⊂ J}. If we write these
maximal cubes as a sequence {Jk }, then Jk is disjoint and k Jk j Ij . By the definition of Jk
in J, we know that |Jk | ≤ Ij ⊂Jk |Ij |. Jensen’s inequality gives |Jk |p ≤ Ij ⊂Jk |Ij |p for 0 < p ≤ 1.
Consequently,
k
|Jk |p ≤
p p
Ij ≤
Ij .
k Ij ⊂Jk
2.7
j
Suppose that E ⊂ j Ij and ξ, ρ ∈ T E. Then ξ ∈ E ⊂ k Jk and ξ ∈ Jk for some
fixed k. For this Jk , we consider the parent dyadic cube Jk , namely, the unique dyadic cube
containing Jk whose sidelength is double of that of Jk . Since Jk is maximal in J, we have that
Jk is not a union of the Ij ’s, that is, Jk contains a point η ∈ Hn \ j Ij ⊂ Hn \ E. Denoting the
boundary of E by ∂E, there is
ρ < distξ, ∂E ≤ diam Jk cn l Jk 2cn lJk .
2.8
If Jk∗ is the cube with the same center as Jk , and 5cn times the sidelength, then
1
dist ξ, ∂Jk∗ ≥ 5cn − 1lJk ≥ 2cn lJk > ρ,
2
since cn > 1,
2.9
which means that ξ, ρ ∈ T Jk∗ . The proof is completed.
Lemma 2.5. Let χI be the characteristic function on the cube I. Then
Hn
q
p
M χI dΛ∞
p ≤ C|I| ,
q > p.
2.10
6
Abstract and Applied Analysis
Proof. Let zI be the center of I. By the definition of the maximal function M, we obtain
⎫
⎧
⎬
⎨
|I|
.
M χI ξ ≤ C inf 1, ⎩ z−1 ξ2n2 ⎭
2.11
I
Since p/q < 1, there is
Hn
q
p
M χI dΛ∞
p ≤ C|I| C
1
|I|p λ−p/q dλ C|I|p .
2.12
0
The proof of Lemma 2.5 is complete.
Lemma 2.6. Let {Ij } be a family of nonoverlapping dyadic cubes. Then there is a maximal subfamily
{Ijk } such that for every dyadic cube I,
p
Ij ≤ 2|I|p ,
k
Ijk ⊂I
Λ∞
p
p
Ij ≤ 2 Ijk .
k
2.13
2.14
Proof. Similar to the proof in 16, if Ij1 I1 , then obviously Ij1 satisfies 2.13. If j1 , . . . , jk have
been chosen so that 2.13 holds, then we define jk1 as the first index such that the family
{Ij1 , . . . , Ijk , Ijk1 } satisfies 2.13. Continuing this proceeding, therefore, we have that {Ijk } is a
maximal subfamily of {Ij } satisfying 2.13. Hence 2.13 holds.
To prove 2.14, let j be an index such that jm < j < jm1 for some m ∈ Z. Then by the
proof of 2.13, there exists a dyadic cube Ij∗ ⊃ Ij such that
p
p p
Ij Ij > 2I ∗ .
k
j
Ijk ⊂Ij∗ ,k≤m
2.15
Therefore,
p
∗
Ij ≤
Ijk ⊂Ij∗ ,k≤m
p
Ij .
k
2.16
We can assume that k |Ijk |p < ∞. Otherwise 2.14 is obviously correct. Then the sequence
{|Ij∗ |} is bounded. Thus, we can consider the family {Il } of maximal cubes of the family {Ij∗ }.
Hence
Ij ⊂
j
Ijk
k
Il .
l
2.17
Abstract and Applied Analysis
7
Consequently, by the definition of Λ∞
p , we obtain
Λ∞
p
p
Ij ≤ 2 Ijk .
2.18
k
The proof is complete.
Proof of Theorem 2.3. By the definition of Λ∞
p , for any ε > 0, there exist {Ij }j such that
p
Ij < Λ∞
p Ek ε,
2.19
j
where Ek {ξ ∈ Hn : 2k < |fξ| ≤ 2k1 }. If Λ∞
p Ek 0, we can choose
Λ∞
p Ek > 0, then
p
Ij ≤ 2Λ∞
p Ek .
j
|Ij |p 0. If
2.20
j
k
By Lemma 2.4, for each integer k, there is a family of nonoverlapping dyadic cubes {Ij }
such that
k
ξ ∈ Hn : 2k < fξ ≤ 2k1 ⊂ Ij ,
j
k p
ξ ∈ Hn : 2k < fξ ≤ 2k1 .
Ij ≤ 2Λ∞
p
2.21
j
k
Set g k 2k1q χFk , where χFk is the characteristic function of Fk j Ij . Then |f|q ≤ g.
First, assume that q > 1. Then the Hölder inequality tells us
!
"
q q
k1q 2
M χI k .
M f ≤M f
≤M g ≤
k
j
j
2.22
Therefore, by Lemma 2.5, we have
Hn
q
M f dΛ∞
2k1q
p ≤ C
j
k
≤C
k
2k1q
Hn
!
"
M χI k dΛ∞
p
j
k p
Ij ≤ C 2k1q Λ∞
p Ek j
k
2kq
22q
q
≤C
Λ∞
ξ ∈ Hn : fξ > λ dλ
p
q −1
2
2k−1q
k
∞
q
q ∞
∞
n f dΛp .
Λp ξ ∈ H : fξ > λ dλ C
≤C
0
Hn
2.23
8
Abstract and Applied Analysis
Now, assume that p < q ≤ 1. Since |f| ≤
k
2k1 χFk , there is
"
!
k1 M f ≤
2
M χI k .
2.24
j
j
k
By using Jensen’s inequality, we obtain
!
"q
q k1q k
M f ≤
2
M χI
.
2.25
j
j
k
Thus, also by Lemma 2.5, there is
Hn
q
M f dΛ∞
2k1q
p ≤ C
j
k
≤C
2k1q
∞
0
Hn
!
"
M χI k dΛ∞
p
j
k p
Ij ≤ C 2k1q Λ∞
p Ek j
k
≤C
Λ∞
p
2.26
k
q
ξ ∈ Hn : fξ > λ dλ C
Hn
q ∞
f dΛp .
That means that 2.5 holds.
For a given λ > 0, let {Ij } be the family of maximal dyadic cubes Ij such that
1
Ij fξdξ > λ.
Note that Mf is dyadic and {z ∈ Hn : Mfz > λ} ⎛
⎞1/p
p
Ij ≤ ⎝ Ij ⎠ ,
j
j
2.27
Ij
j Ij .
Since 0 < p ≤ 1, there is
1/p
∞
Λ∞
.
1 E ≤ Λp E
2.28
Abstract and Applied Analysis
9
Hence
Hn
fzdz ∞
0
≤
1
p
1
p
Λ∞
1
∞
0
∞
0
ξ ∈ Hn : fξ > λ dλ
Λ∞
1
ξ ∈ Hn : fξ > ρ1/p ρ1/p−1 dρ
Λ∞
p
1/p
ξ ∈ Hn : fξ > ρ1/p
ρ1/p−1 dρ
p
1/p−1 ∞ 1 ∞ ∞ Λp ξ ∈ Hn : fξ|p > ρ ρ
Λp ξ ∈ Hn : fξ > ρ dρ
p 0
!
"1/p−1 ∞
p
p
1
f dΛ∞
≤
Λ∞
ξ ∈ Hn : fξ > ρ dρ
p
p
p Hn
0
!
"1/p
p
1
f dΛ∞
,
p
p Hn
2.29
where the last inequality is due to the fact that
Λ∞
p
p
ξ ∈ Hn : fξ > ρ ρ ≤
{ξ∈Hn :|fξ|p >ρ}
p ∞
f dΛp ,
ρ > 0.
2.30
Therefore, by 2.27 and 2.29, we obtain
p
Ij ≤
1 f dξ
λ Ij
p
≤ Cλ
−p
Ij
p ∞
f dΛp .
2.31
For the above {Ij }, by Lemma 2.6 and 2.31, there exists a subfamily {Ijk } such that
⎛
⎞
⎝ Ij ⎠ ≤ 2 Ijk p
Λ∞
z ∈ Hn : M f z > λ Λ∞
p
p
j
≤ Cλ−p
k
k
Ij k
p ∞
f dΛp ≤ Cλ−p
Hn
p ∞
f dΛp .
2.32
The proof of Theorem 2.3 is complete.
3. p-Carleson Measure on Siegel Upper Half Space
Let I be a cube in Hn with center η. The Carleson box based on I is defined by
SI lI
−1 ξ, ρ ∈ U : η ξ ≤
, ρ ≤ lI .
∞
2
n
3.1
10
Abstract and Applied Analysis
For p > 0, a positive Borel measure μ on the Siegel upper half space Un is called a
p-Carleson measure if there exists a constant C > 0 such that
μSI ≤ C|I|p ,
∀ cubes I ⊂ Hn .
3.2
For z ∈ Hn , let Γz {ξ, ρ ∈ Un : |z−1 ξ| < ρ} be the cone at z. Suppose that f is a
measurable function on Un . The nontangential maximal function Nf of f is defined by
N f z sup f ξ, ρ .
3.3
ξ,ρ∈Γz
Since the nontangential maximal function and Hausdorff capacity are defined, we are
paying attention on the characterizations of the p-Carleson measures.
Theorem 3.1. For p ∈ 0, 1, let μ be a positive Borel measure on Un . Then the following three
conclusions are equivalent.
i μ is a p-Carleson measure.
ii There exists a constant C > 0 such that
Un
f ξ, ρ dμ ≤ C
Hn
∞
N f dΛp
3.4
holds for all Borel measurable functions f on Un .
iii For every q > 0, there exists a constant C > 0 such that
Un
q
f ξ, ρ dμ ≤ C
Hn
q ∞
N f dΛp
3.5
holds for all Borel measurable functions f on Un .
Proof. i⇒ii. Assume that μ is a p-Carleson measure, and f is a Borel measurable function
on Un . For λ > 0, let Eλ {z ∈ Hn : Nfz > λ}. If the integral on the right hand side of 3.4
∞
is finite, we may assume that Λp Eλ < ∞. Let {Ij } be any of the dyadic cubes covering of Eλ
with j |Ij |p < ∞. Then Lemma 2.4 tells us that there is a sequence {Jk } of dyadic cubes with
mutually disjointed so that k Jk j Ij , k |Jk |p ≤ j |Ij |p and T Eλ ⊂ k T Jk∗ , where Jk∗
is the cube with the same center and 5cn times sidelength of Jk .
If ξ, ρ ∈ Un satisfies |fξ, ρ| > λ, then Nfz > λ for all z ∈ Bξ, ρ. Thus ξ, ρ ∈
T Eλ , and hence
ξ, ρ ∈ Un : f ξ, ρ > λ ⊂ T Eλ ⊂ T Jk∗ .
k
3.6
Abstract and Applied Analysis
11
By Lemma 2.4, we obtain
μ
T Jk∗
ξ, ρ ∈ U : f ξ, ρ > λ ≤ μ
≤
n
k
' ' p
≤ C'μ' Jj∗ .
' ' p
μ T Jk∗ ≤ C'μ' Jk∗ k
k
3.7
j
Taking an infimum over all such dyadic cube coverings, we have
μ
' ' ∞
ξ, ρ ∈ Un : f ξ, ρ > λ ≤ C'μ'Λp Eλ .
3.8
Therefore,
Un
f ξ, ρ dμ ∞
μ
ξ, ρ ∈ Un : f ξ, ρ > λ dλ
0
≤C
∞
' '
' ' ∞
'μ'Λp Eλ dλ C'μ'
0
Hn
∞
N f dΛp .
3.9
ii⇒i. Suppose that 3.4 is valid for all Borel measurable functions on Un . For a cube
I ⊂ H , let φξ, ρ χT I . Note that ξ, ρ ∈ T I ∩ Γz if and only if z ∈ Bξ, ρ ⊂ I, there is
Nφ χI . Then, by 3.4, we have
n
μT I ≤ C
∞
∞ Λp
N φ > λ dλ C
0
1
∞
Λp Idλ ≤ C|I|p .
3.10
0
It means that μ is a p-Carleson measure.
ii⇒iii. Replacing |f| with |f|q in 3.4 for q > 0, we immediately obtain
Un
q
f ξ, ρ dμ ≤ C
Hn
q ∞
N f dΛp .
3.11
iii⇒ii. If we set q 1 in 3.5, then 3.4 holds.
The proof of Theorem 3.1 is complete.
To continue the characterization of the p-Carleson measures on the Siegel upper
half space, we need to prove the following Lemma 3.2. It is said that the nontangential
maximal functions are dominated by the dyadic Hardy-Littlewood maximal operators on
the Heisenberg group.
12
Abstract and Applied Analysis
Lemma 3.2. Let f be a locally
integrable function on Hn , and φ ∈ SHn be a nonnegative radial and
(
decreasing function with Hn φz dz 1. Then
NFz sup F ξ, ρ ξ,ρ∈Γz
sup f ∗ φρ ξ ≤ CM f z,
ξ,ρ∈Γz
3.12
where φρ ξ ρ−2n2 φρ−1 ξ and ρ > 0 is real.
Proof. Since φ ∈ SHn , we can suppose that |φz| ≤ C/1 |z|2n2ε for some ε > 0. Then
f ∗ φρ ξ ≤
Hn
ρ−2n2 φ ρ−1 η−1 ξ f η dη
|η−1 ξ|≤ρ
∞ 2k ρ<|η−1 ξ|≤2k1 ρ
k0
≤
ρ−2n2 φ ρ−1 η−1 ξ f η dη
C
ρ2n2
|η−1 ξ|≤ρ
ρ−2n2 φ ρ−1 η−1 ξ f η dη
∞
f η dη C
k0
∞
1
≤ CM f ξ C
2n2
ρ
k0
2k ρ<|η−1 ξ|≤2k1 ρ
|η−1 ξ|≤2k1 ρ
1
2k2n2ε
ρ−2n2 f η 2n2ε dη
1 η−1 ξ /ρ
3.13
f η dη
∞
≤ C M f ξ 2−k1ε C M f ξ.
k0
Hence
NFz sup f ∗ φρ ξ ≤ CM f z.
ξ,ρ∈Γz
3.14
Lemma 3.2 is proved.
Theorem 3.3. For 0 < p ≤ 1, q > p, let μ be a positive Borel measure on Un . Then μ is a p-Carleson
measure if and only if
Un
q
G ξ, ρ dμ ≤ C
Hn
q ∞
g dΛp
3.15
holds for all functions G on Un which can be written as Gξ, ρ g ∗ φρ ξ, where g is a locally
integrable function on Hn , and φ is the function in Lemma 3.2.
Abstract and Applied Analysis
13
Proof. Suppose that μ is a p-Carleson measure. By the conclusions of Theorems 3.1 and 2.3
and Lemma 3.2, there is
Un
q
G ξ, ρ dμ ≤ C
∞
Hn
NGq dΛp
≤C
q ∞
M g dΛp ≤ C
Hn
Hn
q ∞
g dΛp . 3.16
Conversely, for any cube I ⊂ Hn , if we set g χI , then for ξ, ρ ∈ T I, there is
G ξ, ρ −1
φρ η ξ dη ≥
−1
φρ η ξ dη I
Bξ,ρ
φ1 η−1 ξ dη
Bξ,1
3.17
φzdz c.
B0,1
Thus, by using the inequality 3.15, we obtain
cq μT I cq
C
T I
Hn
dμ ≤
∞
χI dΛp
T I
q
G ξ, ρ dμ ≤
∞
CΛp I
Un
q
G ξ, ρ dμ ≤ C
Hn
q ∞
g dΛp
3.18
p
≤ C|I| .
This ends the proof of Theorem 3.3.
∞
Let NΛp be the space of all Borel measurable functions f on Un satisfying
' '
'f ' ∞ NΛp Hn
∞
N f dΛp < ∞.
3.19
∞
gives a quasi norm and NΛp is complete.
Then · NΛ∞
p ∞
The following result says that NΛp is the dual of p-Carleson measure.
Theorem 3.4. Let p ∈ 0, 1. Then there exists a duality between the space of p-Carleson measures
∞
and NΛp in the following sense.
∞
i Every p-Carleson measure μ on Un defines a bounded linear functional on NΛp via the
pairing
)
∞
*
μ, f Un
f ξ, ρ dμ.
3.20
∞
ii Let N0 Λp be the closure in NΛp of the continuous functions with compact support
∞
in Un . Then every bounded linear functional on N0 Λp given via the pairing 3.20 by a
Borel measure μ on Un is a p-Carleson measure.
14
Abstract and Applied Analysis
∞
Proof. Assume that μ is a p-Carleson measure and f ∈ NΛp . By 3.19 and ii in
Theorem 3.1, we have
Un
f ξ, ρ dμ ≤ C
Hn
∞
N f dΛp < ∞.
3.21
(
Thus, μ, f Un fξ, ρdμ is well defined. Hence, every p-Carleson measure μ defines a
∞
bounded linear functional on NΛp . The part i is proved.
∞
For part ii, let L be a bounded linear functional on N0 Λp . By the continuity and
∞
the closure of N0 Λp , applying the Riesz representation theorem, we obtain a Borel measure
μ on Un having
L f Un
)
*
f ξ, ρ dμ μ, f .
3.22
If f χT I for any cube I ∈ Hn , then
μT I Un
L
'
'
χT I dμ L χT I ≤ L · 'χT I 'NΛ∞
p Hn
∞
N χT I dΛp CL
3.23
∞
dΛp
∞
CLΛp I
p
≤ CL · |I| .
I
Hence, μ is a p-Carleson measure with μ ≤ CL. This ends the proof of Theorem 3.4.
4. Tent Spaces with Hausdorff Capacity
With Hausdorff capacity on the Heisenberg group discussed above, in this section, we
introduce the tent spaces on the Siegel upper half space, an analogy of the Coifman-MeyerStein tent space on Rn cf. 4, 6. Then the atomic decomposition of the tent spaces and the
duality of the tent space are discussed.
Definition 4.1. Let p ∈ 0, 1. A Lebesgue measurable function f on Un is said to belong to Tp∞
if
' '
'f ' ∞ sup
Tp
B⊂Hn
1
|B|p
T B
2
f ξ, ρ dξdρ
1/2
< ∞,
ρ12n11−p
4.1
where B runs over all balls in Hn , and T B is a tent based on B.
Definition 4.2. Let p ∈ 0, 1. The tent space Tp1 consists of all measurable functions f on Un for
which
' '
'f ' 1 inf
Tp
w
Un
2 −1
f ξ, ρ w ξ, ρ
dξdρ
ρ1−2n11−p
1/2
< ∞,
4.2
Abstract and Applied Analysis
15
where the infimum is taken over all nonnegative Borel measurable functions w on Un satisfying
∞
Hn
NwdΛp
≤ 1,
4.3
and w is allowed to vanish only where f vanishes.
We will identify Tp∞ with a dual space of Tp1 . In order to do this, we first introduce the
1
Tp -atom as follows.
Definition 4.3. A function a on Un is said to be a Tp1 -atom, if there exists a ball B ⊂ Hn such
that a is supported in the tent T B and satisfies
T B
2
a ξ, ρ dξdρ
ρ1−2n11−p
≤
1
.
|B|p
4.4
For the tent space Tp1 on Un and · Tp1 , we have the triangle inequality with a constant
in the following lemma.
Lemma 4.4. Let p ∈ 0, 1. If
j
gj Tp1 < ∞, then g j
gj ∈ Tp1 and
' '
' '
'g ' 1 ≤ C 'gj ' 1 .
Tp
T
4.5
p
j
Proof. Without loss of generality, we assume that λj gj Tp1 > 0 for all j. Set fj g−1
g.
Tp1 j
(
∞
Then fj Tp1 ≤ 1 and g j λj fj . Suppose that wj ≥ 0 for all j such that Hn Nwj dΛp ≤ 1
and
Un
2 −1
gj ξ, ρ wj ξ, ρ
dξdρ
ρ1−2n11−p
' '2
≤ 2'gj 'T 1 .
p
4.6
According to the definition of Tp1 , the above inequality holds obviously. By using the CauchySchwarz inequality, we have
⎛
⎞⎛
⎞
2
2
g ≤ ⎝ λj wj ⎠⎝ λj fj w−1 ⎠.
j
j
4.7
j
Let w j λj −1 j λj wj . Notice that the vanishing of w implies the vanishing of all wj ,
which can only happen whenever all the gj vanish, that is, g 0, then
∞
Hn
NwdΛp
⎛
⎞−1
≤ ⎝ λj ⎠
λj
j
j
Hn
∞
N wj dΛp ≤ 1.
4.8
16
Abstract and Applied Analysis
Thus, by the inequality 4.7, we obtain
Un
2 −1
g ξ, ρ w ξ, ρ
dξdρ
ρ1−2n11−p
⎛
⎞
≤ C⎝ λj ⎠ λj
j
⎛
≤ 2C⎝
j
j
Hn
2 −1
fj ξ, ρ wj ξ, ρ
dξdρ
4.9
ρ1−2n11−p
⎞
⎞2
⎞2
⎛
⎛
' '2
' '
λj ⎠ λj 'fj 'T 1 C⎝ λj ⎠ C⎝ 'gj 'T 1 ⎠ .
p
j
j
j
p
Taking the infimum on the left above inequality, we have
' '
' '
'g ' 1 ≤ C 'gj ' 1 .
Tp
T
j
4.10
p
The proof of Lemma 4.4 is complete.
Remark 4.5. By Lemma 4.4, one can show that · Tp1 is a quasinorm and the tent space Tp1 is
complete.
The main result of this section is the atomic decomposition of the tent space Tp1 as
follows.
Theorem 4.6. Let p ∈ 0, 1. Then a function f on Un belongs to Tp1 if and only if there exist a
sequence of Tp1 -atoms {aj } and an l1 -sequence {λj } such that
f
λj aj .
4.11
j
Moreover,
' '
'f '
Tp1
⎧
⎫
⎨ ⎬
λj : f ≈ inf
λj aj ,
⎩ j
⎭
j
4.12
where the infimum is taken over all possible atomic decompositions of f ∈ Tp1 .
Note that the right hand side of 4.12 in fact defines a norm and then Tp1 becomes a
Banach space.
Proof. Suppose that a is a Tp1 -atom on Un . Then there exists a ball B Bz, r ⊂ Hn such that
supp a ⊂ T B and
T B
2
a ξ, ρ dξdρ
ρ1−2n11−p
≤
1
.
|B|p
4.13
Abstract and Applied Analysis
17
Fix ε > 0, and let
+ !
−2n2p
w ξ, ρ hr
min 1,
r
Dξ, z
"2n2pε ,
,
4.14
where Dξ, z denotes the distance between ξ, ρ and z, 0 in Un , and h is a suitable constant
which will be chosen later. Obviously, wξ, ρ is identically equal to hr −2n2p on the upper
half ball of radius r center is z, 0, and Dξ, z ≤ r and decays radially outside the ball
outside the ball Dξ, z > r, and Dξ, z → ∞. For ξ ∈ Hn , the distance in Un from the cone
Γξ to z, 0 is c|z−1 ξ| cdz, ξ. Then
Nwz ≤ hr
−2n2p
+ !
min 1,
r
cdξ, z
"2n2pε ,
.
4.15
Therefore,
−1
h
Hn
∞
NwdΛp
∞
∞
Λp
ξ ∈ Hn : h−1 Nw > λ dλ
0
≤
r −2n2p
- ∞
Λp
0
≤C
r −2n2p
.
! "
1 r ε 1/2n2pε
dλ
B z,
c λ
- ∞
Λ
p
0
.
! "
1 r ε 1/2n2pε
dλ
B z,
c λ
4.16
2n2p r −2n2p
≤ C r ε/2n2pε
λ−2n2p/2n2pε dλ C.
0
(
∞
Thus, Hn Nw dΛp ≤ 1 by choosing h C−1 . On the other hand, let w−1 r 2n2p on T B.
We have
T B
2 −1
a ξ, ρ w ξ, ρ
dξdρ
ρ1−2n11−p
r
2n2p
≤r
2n2p
T B
−p
|B|
2
a ξ, ρ dξdρ
ρ1−2n11−p
4.17
C.
Therefore, a ∈ Tp1 and aTp1 ≤ C.
Now, taking a sum j λj aj , where j |λj | < ∞ and every aj is Tp1 -atom on Un , by
Lemma 4.4, the sum converges in the quasinorm to f ∈ Tp1 with
' '
' '
'f ' 1 ≤
λj 'aj ' 1 ≤ C λj < ∞.
Tp
T
j
That means f j
λj aj ∈ Tp1 .
p
j
4.18
18
Abstract and Applied Analysis
Conversely, let f ∈ Tp1 . We choose a Borel measurable function w ≥ 0 on Un such that
4.3 holds and
Un
∞
Λp ,
2 −1
f ξ, ρ w ξ, ρ
dξdρ
ρ1−2n11−p
' '2
≤ 2'f 'Tp1 .
4.19
For each k ∈ Z, let Ek {Z ∈ Hn : Nwz > 2k }. By Lemma 2.4 and the definition of
it follows that there exists a disjoint dyadic cubes sequence {Ik,j } such that
p
Ik,j ≤ 2Λ∞
p Ek ,
T Ek ⊂
j
S∗ Ik,j ,
j
4.20
where S∗ Ik,j {ξ, ρ ∈ Un : ξ ∈ Ik,j , ρ < 2 diamIk,j }. Define
∗
Tk,j S∗ Ik,j \
S Im,l .
4.21
m>k l
Then Tk,j is disjoint in Un for different j, k. Therefore,
K ∗
Tk,j S∗ I−K,j \
S Im,l ⊃ T E−K \
S∗ Im,l .
k−K j
j
m>k l
4.22
m>k l
Note that k T Ek {ξ, ρ ∈ Un : wξ, ρ > 0}, ξ, ρ ∈
/ T Ek implies wξ, ρ ≤ 2k . And each
S∗ Im,l is contained in a cube of sidelength 4 diamIm,l in Un . By using the subadditivity of
the p-Hausdorff capacity and 4.3, there is
∞
Λp
S∗ Im,l m>k l
≤C
|Im,l |p ≤ C
m>k l
∞
Λp Em −→ 0
4.23
m>k
as k → ∞. Thus, by 4.21,
k
j
Tk,j ⊃
k
T Ek \
/
k m>k l
S∗ Im,l ξ, ρ ∈ Un : w ξ, ρ > 0 \ E∞ ,
4.24
Abstract and Applied Analysis
19
where E∞ is a set of zero p-Hausdorff capacity, that is,
∞
Λp E∞ /
S∗ Im,l ∞
Λp
0.
4.25
k m>k l
Since w is allowed to vanish only where f vanishes, we have f p ∗ Ik,j ak,j 2
f ξ, ρ Tk,j
λk,j p ∗ Ik,j ρ1−2n11−p
Tk,j
k,j
f · χTk,j a.e. on Un . Let
−1/2
dξdρ
2
f ξ, ρ dξdρ
f · χTk,j ,
4.26
1/2
ρ1−2n11−p
,
∗
∗
where the Ik,j
is a cube as Lemma 2.4, that is, Ik,j
5cn Ik,j .
Thus
f
λk,j ak,j ,
a.e. on Un .
4.27
k,j
Note that, by 4.21, Tk,j ⊂ S∗ Ik,j ⊂ T Bk,j , where Bk,j is the ball with the same center as Ik,j
∗
and radius clIk,j
c > 1 is a constant. Thus, ak,j is supported in T Bk,j , and
T Bk,j 2
ak,j ξ, ρ dξdρ
ρ1−2n11−p
p ∗ Ik,j 2
f ξ, ρ Tk,j
−1
dξdρ
ρ1−2n11−p
T Bk,j 2
f · χTk,j dξdρ
4.28
ρ1−2n11−p
−p
−p
∗ Ik,j
CBk,j .
This means that every ak,j is a Tp1 -atom. The remaining is to estimate
Notice the fact that
w≤2
k1
on Tk,j ⊂
S∗ Ik1,l l
k,j
|λk,j |.
c
⊂ T Ek1 c .
4.29
20
Abstract and Applied Analysis
By the Cauchy-Schwarz inequality, 4.19 and 4.20, we have
p/2 k1/2
∗
λk,j ≤
2
Ik,j k,j
k,j
2 −1
f ξ, ρ w ξ, ρ
Tk,j
⎞1/2 ⎛
⎛
p
k1 ∗
⎠
⎝
⎝
≤
2 Ik,j k,j
k,j
1/2
dξdρ
ρ1−2n11−p
2 −1
f ξ, ρ w ξ, ρ
Tk,j
⎞1/2
dξdρ
ρ1−2n11−p
⎛
⎞1/2
p
' '
' '
∞
k
Ik,j ⎠ ≤ C'f ' 1
≤ C'f 'Tp1 ⎝ 2
2k Λp Ek Tp
j
k
' '
C'f 'Tp1
' '
≤ C'f 'Tp1
If f k
! ∞
⎠
1/2
4.30
k
"1/2
∞
Λp {ξ ∈ Hn : Nw > λ}dλ
0
!
∞
Hn
"1/2
NwdΛp
' '
≤ C'f 'Tp1 .
λk ak and every ak is a Tp1 -atom, then, by Lemma 4.4, we obtain
' '
'f ' 1 ≤ C |λk |ak 1 ≤ C |λk |.
T
Tp
p
k
k
4.31
Thus
' '
'f '
Tp1
⎧
⎫
⎨ ⎬
λj : f ≈ inf
λj aj ,
⎩ j
⎭
j
4.32
where the infimum is taken over all atomic decompositions of f ∈ Tp1 . The proof is complete.
The dual result is as follows.
Theorem 4.7. Let p ∈ 0, 1. Then the dual of Tp1 can be identified with Tp∞ under the pairing
)
*
f, g dξdρ
.
f ξ, ρ g ξ, ρ
ρ
Un
4.33
Proof. We first show that
' ' ' '
dξdρ
f ξ, ρ g ξ, ρ ≤ C'f 'Tp1 'g 'Tp∞
ρ
Un
holds for all f ∈ Tp1 and g ∈ Tp∞ .
4.34
Abstract and Applied Analysis
21
In fact, assume that w is a nonnegative Borel measurable function on Un satisfying
inequality 4.3 in Definition 4.2 and g ∈ Tp∞ . Then there exists a constant C for any ball
B ⊂ Hn satisfying
1
|B|p
T B
2
g ξ, ρ dξdρ
ρ12n11−p
≤ C.
4.35
It means that |gξ, ρ|2 ρ−2n11−p−1 dξdρ is a p-Carleson measure, and μ ≈ g2Tp∞ . Hence,
by Theorem 3.1, we obtain
Un
' '2
2
w ξ, ρ g ξ, ρ ρ−2n11−p−1 dξdρ ≤ C'g 'Tp∞
∞
Hn
NwdΛp
' '2
≤ C'g 'Tp∞ .
4.36
Thus, for f ∈ Tp1 , by using the Cauchy-Schwarz inequality, there is
dξdρ
f ξ, ρ g ξ, ρ ρ
Un
2 −1
dξdρ
f w
≤
1−2n11−p
ρ
Un
≤C
Un
2 −1
f ξ, ρ w ξ, ρ
1/2 Un
2
g w
dξdρ
ρ12n11−p
1/2
dξdρ
ρ1−2n11−p
1/2
4.37
' '
· 'g 'Tp∞
' ' ' '
≤ C'f 'Tp1 'g 'Tp∞ .
This gives 4.34. Thus, every g ∈ Tp∞ induces a bounded linear functional on Tp1 via the
pairing 4.33. It suffices to prove the converse.
Let L be a bounded linear functional on Tp1 and fix a ball B Bz, r ⊂ Hn . If f is
supported in T B with f ∈ L2 T B, ρ−1 dξdρ, then
T B
2
f ξ, ρ dξdρ
ρ1−2n11−p
≤ r 2n11−p
Cr
2n1
2 dξdρ
f ξ, ρ ρ
T B
' '2
|B| f 'L2 T B,ρ−1 dξdρ .
4.38
−p '
Therefore, f is a multiple of a Tp1 -atom with
' '2
' '
'f ' 1 ≤ Cr 2n1 'f '2 2
.
Tp
L T B,ρ−1 dξdρ
4.39
22
Abstract and Applied Analysis
Hence, L induces a bounded linear functional on L2 T B, ρ−1 dξdρ. Thus, there exists a
function g which is locally in L2 Un , ρ−1 dξdρ such that
L f dξdρ
,
f ξ, ρ g ξ, ρ
ρ
Un
4.40
whenever f ∈ Tp1 with support in some finite tent T B. By the atomic decomposition of tent
function in Theorem 4.6, obviously, the subspace of such f is dense in Tp1 . Therefore, the rest
of the proof is to show that g ∈ Tp∞ and gTp∞ ≤ CL.
Now, again fix a ball B ⊂ Hn and for every ε > 0, let
fε ξ, ρ ρ−2n11−p g ξ, ρ χT ε B ξ, ρ ,
4.41
where T ε B T B ∩ {ξ, ρ : ρ > ε} is the truncated of T B.
Since g ∈ L2 T B, there is
T B
2
fε ξ, ρ dξdρ
ρ1−2n11−p
T ε B
2
g ξ, ρ dξdρ
ρ12n11−p
< ∞.
4.42
Hence, fε is a multiple of a Tp1 -atom with
' '2
'fε ' 1 ≤ Cr 2n1p
Tp
T ε B
2
g ξ, ρ dξdρ
ρ12n11−p
4.43
,
where C is independent of ε. By 4.40, we obtain
T ε B
2
g ξ, ρ dξdρ
ρ12n11−p
' '
L fε ≤ L · 'fε 'Tp1
≤ CL |B|
p
T ε B
2
g ξ, ρ dξdρ
ρ12n11−p
4.44
1/2
.
Thus
−p
|B|
T ε B
2
g ξ, ρ dξdρ
1/2
ρ12n11−p
≤ CL.
4.45
≤ CL,
4.46
It follows that
−p
|B|
T B
2
g ξ, ρ dξdρ
ρ12n11−p
1/2
Abstract and Applied Analysis
23
since 4.40 is true for all ε > 0. Therefore, g ∈ Tp∞ and gTp∞ ≤ CL. In fact, in 4.40, we
can replace the local function g by ordinary. Thus, we obtain the representation of L via the
pairing 4.40 for all f ∈ Tp1 . This ends the proof of Theorem 4.7.
Acknowledgments
The author would like to thank the referees for their meticulous work and helpful suggestions, which improve the presentation of this paper. This work was supported by the NNSF
of China no. 11041004 and the NSF of Shandong province of China no. ZR2010AM032.
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International Journal of
Advances in
Combinatorics
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Mathematical Physics
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Volume 2014
Journal of
Complex Analysis
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Volume 2014
International
Journal of
Mathematics and
Mathematical
Sciences
Journal of
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Stochastic Analysis
Abstract and
Applied Analysis
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International Journal of
Mathematics
Volume 2014
Volume 2014
Discrete Dynamics in
Nature and Society
Volume 2014
Volume 2014
Journal of
Journal of
Discrete Mathematics
Journal of
Volume 2014
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Applied Mathematics
Journal of
Function Spaces
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Volume 2014
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Volume 2014
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Volume 2014
Optimization
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Volume 2014
Hindawi Publishing Corporation
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Volume 2014