# Carleson Measure and Tent Spaces on the Siegel Upper Half Space

Authors: Kai Zhao
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

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.


---

## Body

## 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 onℝn 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, 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 on the Choquet integral space LΛp∞q(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.Letℤ, ℝ, and ℂ be the sets of all integers, real numbers, and complex numbers, and ℤn, ℝn, and ℂn be n-dimensional ℤ, ℝ, and ℂ, respectively. The Siegel upper half space 𝕌n in ℂn+1 is defined by
(1.1)𝕌n={z=(z′,zn+1)∈ℂn+1:Imzn+1>|z′|2},
where z'=(z1,z2,…,zn)∈Cn,|z'|2=∑k=1n‍|zk|2. The boundary of 𝕌n is
(1.2)∂𝕌n={z=(z′,zn+1)∈ℂn+1:Imzn+1=|z′|2}.The Heisenberg group onℂn×ℝ, denoted by ℍn, is a noncommutative nilpotent Lie group with the underlying manifold ℂn×ℝ. The group law is given by
(1.3)zw=(z′,t)(w′,s)=(z′+w′,t+s+2Imz′w′¯),forz=(z′,t),w=(w′,s)∈ℍn,
where z'w'¯=z1w1¯+⋯+znwn¯is the Hermitean product on ℂn.It is easy to check that the inverse of the elementz=(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 elementζ=(ζ',t) of ℍn, the following affine self-mapping of 𝕌n:
(1.4)(ζ′,t):(z′,zn+1)↦(z′+ζ′,zn+1+t+2iz′ζ′¯+i|ζ′|2)
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]).The dilations onℍn are defined by δz=δ(z',t)=(δz',δ2t), δ>0, and the rotation on ℍn is defined by σz=σ(z',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
(1.5)|z|=(|z′|4+|t|2)1/4,z=(z′,t)∈ℍn,
which is homogeneous of degree 1 and satisfies |z-1|=|z| and |zw|≤C(|z|+|w|) for some absolute constant C. The distance function d(z,w) of point z and w in ℍn is defined by d(z,w)=|w-1z|.Forz=(z',t)∈ℍn (or z=(x,y,t)∈ℍn for x=(x1,x2,,…,xn), y=(y1,y2,,…,yn)), we define
(1.6)|z|∞=max{|x1|,…,|xn|,|y1|,…,|yn|,|t|},
where z'=(z1,…,zn), and zk=xk+iyk,1≤k≤n.To define the cube inℍn, 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 F(x,y) by
(1.7)F(x,y)=∑j=1∞‍24j(〈2jx〉([2jy]mod(2ℤn))-〈2jy〉([2jx]mod(2ℤn))),
and set
(1.8)I0={(x,y,t)∈ℍn:0≤xk≤1,0≤yk≤1,1≤k≤n,0≤t-F(x,y)≤1},Γ={(m,n,l)∈ℍn:m,n∈ℤn;l∈ℤ}. Then
(1.9)ℍn=∑γ∈Γ‍γ-1I0′,(disjointunion),
where I0′={(x,y,t)∈ℍn:0≤xk<1,0≤yk<1,k=1,2,…,n,0≤t-F(x,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-1I0. Let |I| (with the Lebesgue measure) be the volume of I with length l(I). It is easy to see that |I|=[l(I)]2n+2. Obviously, the diameter of I denoted by diam(I) is equal to cnl(I), where cn is a constant depending only on n. The “dyadic cubes" on ℍn can be defined by I=2-jγ-1I0, γ∈Γ, j∈ℤ. We also call the “cube" and “dyadic cube" as cube and dyadic cube, respectively.A ball inℍn with center w and radius r is denoted as B=B(w,r)={z:|w-1z|<r}. The tent based on the set E⊂ℍn is defined by
(1.10)T(E)={(ξ,ρ)∈𝕌n:B(ξ,ρ)⊂E}.The Schwartz class of rapidly decreasing smooth function onℍn will be denoted by 𝒮(ℍn). The dual of 𝒮(ℍn) is 𝒮'(ℍn), the space of tempered distributions on ℍn.


## 2. Hausdorff Capacity on Heisenberg Group

In order to discuss the fractional Carleson measures on the Siegel upper half space, we introduce thep-dimensional Hausdorff capacity and Choquet integral on the Heisenberg group in the following.Definition 2.1.
Forp∈(0,1] and E⊂ℍn, the p-dimensional Hausdorff capacity of E is defined by
(2.1)Λp∞(E)=inf{∑j‍|Ij|p:E⊂⋃j=1∞‍Ij},
where the infimum ranges only over covers of E by dyadic cubes.Remark 2.2.
It is easy to check thatΛp∞ satisfies the following properties.(i)
IfEj is nondecreasing, then Λp∞(⋃j‍Ej)=limj→∞Λp∞(Ej).(ii)
IfEj is nonincreasing, then Λp∞(⋂j‍Ej)=limj→∞Λp∞(Ej).(iii)
Λp∞ has the strong subadditivity condition. 										(2.2)Λp∞(E1⋃E2)+Λp∞(E1∩E2)≤Λp∞(E1)+Λp∞(E2).For a nonnegative functionf on ℍn, the Choquet integral with respect to the Hausdorff capacity on the Heisenberg group is defined by
(2.3)∫ℍn‍f(z)dΛp∞=∫0∞‍Λp∞({z∈ℍn:f(z)>λ})dλ.SinceΛp∞ 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 Section3, we need to discuss the Hardy-Littlewood maximal operator on the Heisenberg group.Letf∈Lloc1(ℍn). M(f), the dyadic Hardy-Littlewood maximal operator of f, is defined by
(2.4)M(f)(z)=supI∋z1|I|∫I‍|f(ξ)|dξ,
where the supremum is taken over all dyadic cubes I containing z.The boundedness of the dyadic Hardy-Littlewood maximal operators on the Choquet integral spacesLΛp∞q(ℍn) is as follows.Theorem 2.3.
Suppose0<p≤1. There exists a constant C depending only on p and n such that
(2.5)∫ℍn‍M(f)qdΛp∞≤C∫ℍn‍|f|qdΛp∞,∀f∈LΛp∞q(ℍn),q>p,(2.6)Λp∞({ξ∈ℍn:M(f)(ξ)>λ})≤Cλp∫ℍn‍|f|pdΛp∞,∀f∈LΛp∞p(ℍn).To prove Theorem2.3, we need to prove the following lemmas first.Lemma 2.4.
Forp∈(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 ⋃kJk=⋃jIj and ∑k‍|Jk|p≤∑j‍|Ij|p. Moreover, if E⊂⋃jIj, then the following tent inclusion T(E)⊂⋃k‍T(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 ⋃kJk=⋃jIj. 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,
(2.7)∑k‍|Jk|p≤∑k‍∑Ij⊂Jk‍|Ij|p≤∑j‍|Ij|p.
Suppose thatE⊂⋃jIj and (ξ,ρ)∈T(E). Then ξ∈E⊂⋃kJk 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∖(⋃jIj)⊂ℍn∖E. Denoting the boundary of E by ∂E, there is
(2.8)ρ<dist(ξ,∂E)≤diam(Jk′)=cnl(Jk′)=2cnl(Jk).
If Jk* is the cube with the same center as Jk, and 5cn times the sidelength, then
(2.9)dist(ξ,∂Jk*)≥12(5cn-1)l(Jk)≥2cnl(Jk)>ρ,(sincecn>1),
which means that (ξ,ρ)∈T(Jk*). The proof is completed.Lemma 2.5.
LetχI be the characteristic function on the cube I. Then
(2.10)∫ℍn‍M(χI)qdΛp∞≤C|I|p,q>p.Proof.
LetzI be the center of I. By the definition of the maximal function M, we obtain
(2.11)M(χI)(ξ)≤Cinf{1,|I||zI-1ξ|(2n+2)}.
Since p/q<1, there is
(2.12)∫ℍn‍M(χI)qdΛp∞≤C|I|p+C∫01‍|I|pλ-p/qdλ=C|I|p.
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,
(2.13)∑Ijk⊂I‍|Ijk|p≤2|I|p,(2.14)Λp∞(⋃Ij)≤2∑k‍|Ijk|p.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 jk+1 as the first index such that the family {Ij1,…,Ijk,Ijk+1} 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<jm+1 for some m∈ℤ. Then by the proof of (2.13), there exists a dyadic cube Ij*⊃Ij such that
(2.15)∑Ijk⊂Ij*,k≤m‍|Ijk|p+|Ij|p>2|Ij*|p.
Therefore,
(2.16)|Ij*|p≤∑Ijk⊂Ij*,k≤m‍|Ijk|p.
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 {I~l} of maximal cubes of the family {Ij*}. Hence
(2.17)⋃j‍Ij⊂(⋃k‍Ijk)⋃(⋃l‍I~l).
Consequently, by the definition of Λp∞, we obtain
(2.18)Λp∞(⋃Ij)≤2∑k‍|Ijk|p.
The proof is complete.Proof of Theorem2.3.
By the definition ofΛp∞, for any ε>0, there exist {Ij}j such that
(2.19)∑j‍|Ij|p<Λp∞(Ek)+ε,
where Ek={ξ∈ℍn:2k<|f(ξ)|≤2k+1}. If Λp∞(Ek)=0, we can choose ∑j‍|Ij|p=0. If Λp∞(Ek)>0, then
(2.20)∑j‍|Ij|p≤2Λp∞(Ek).
By Lemma 2.4, for each integer k, there is a family of nonoverlapping dyadic cubes {Ij(k)} such that
(2.21){ξ∈ℍn:2k<|f(ξ)|≤2k+1}⊂⋃j‍Ij(k),∑j‍|Ij(k)|p≤2Λp∞({ξ∈ℍn:2k<|f(ξ)|≤2k+1}).
Set g=∑k‍2(k+1)qχFk, where χFk is the characteristic function of Fk=⋃jIj(k). Then |f|q≤g.
First, assume thatq>1. Then the Hölder inequality tells us
(2.22)M(f)q≤M(|f|q)≤M(g)≤∑k‍2(k+1)q∑j‍M(χIj(k)).
Therefore, by Lemma 2.5, we have
(2.23)∫ℍn‍M(f)qdΛp∞≤C∑k‍2(k+1)q∑j‍∫ℍn‍M(χIj(k))dΛp∞≤C∑k‍2(k+1)q∑j‍|Ij(k)|p≤C∑k‍2(k+1)qΛp∞(Ek)≤C∑k‍22q2q-1∫2(k-1)q2kq‍Λp∞({ξ∈ℍn:|f(ξ)|q>λ})dλ≤C∫0∞‍Λp∞({ξ∈ℍn:|f(ξ)|q>λ})dλ=C∫ℍn‍|f|qdΛp∞.
Now, assume that p<q≤1. Since |f|≤∑k‍2k+1χFk, there is
(2.24)M(f)≤∑k‍2k+1∑j‍M(χIj(k)).
By using Jensen’s inequality, we obtain
(2.25)M(f)q≤∑k‍2(k+1)q∑j‍M(χIj(k))q.
Thus, also by Lemma 2.5, there is
(2.26)∫ℍn‍M(f)qdΛp∞≤C∑k‍2(k+1)q∑j‍∫ℍn‍M(χIj(k))dΛp∞≤C∑k‍2(k+1)q∑j‍|Ij(k)|p≤C∑k‍2(k+1)qΛp∞(Ek)≤C∫0∞‍Λp∞({ξ∈ℍn:|f(ξ)|q>λ})dλ=C∫ℍn‍|f|qdΛp∞.
That means that (2.5) holds.
For a givenλ>0, let {Ij} be the family of maximal dyadic cubes Ij such that
(2.27)1|Ij|∫Ij‍|f(ξ)|dξ>λ.
Note that M(f) is dyadic and {z∈ℍn:M(f)(z)>λ}=⋃j‍Ij. Since 0<p≤1, there is
(2.28)∑j‍|Ij|≤(∑j‍|Ij|p)1/p,Λ1∞(E)≤Λp∞(E)1/p.
Hence
(2.29)∫ℍn‍|f(z)|dz=∫0∞‍Λ1∞({ξ∈ℍn:|f(ξ)|>λ})dλ=1p∫0∞‍Λ1∞({ξ∈ℍn:|f(ξ)|>ρ1/p})ρ(1/p)-1dρ≤1p∫0∞‍Λp∞({ξ∈ℍn:|f(ξ)|>ρ1/p})1/pρ(1/p)-1dρ=1p∫0∞‍[Λp∞({ξ∈ℍn:f(ξ)|p>ρ})ρ](1/p)-1Λp∞({ξ∈ℍn:|f(ξ)|p>ρ})dρ≤1p(∫ℍn‍|f|pdΛp∞)(1/p)-1∫0∞‍Λp∞({ξ∈ℍn:|f(ξ)|p>ρ})dρ=1p(∫ℍn‍|f|pdΛp∞)1/p,
where the last inequality is due to the fact that
(2.30)Λp∞({ξ∈ℍn:|f(ξ)|p>ρ})ρ≤∫{ξ∈ℍn:|f(ξ)|p>ρ}‍|f|pdΛp∞,ρ>0.
Therefore, by (2.27) and (2.29), we obtain
(2.31)|Ij|p≤(1λ∫Ij‍|f|dξ)p≤Cλ-p∫Ij‍|f|pdΛp∞.
For the above {Ij}, by Lemma 2.6 and (2.31), there exists a subfamily {Ijk} such that
(2.32)Λp∞({z∈ℍn:M(f)(z)>λ})=Λp∞(⋃j‍Ij)≤2∑k‍|Ijk|p≤Cλ-p∑k‍∫Ijk‍|f|pdΛp∞≤Cλ-p∫ℍn‍|f|pdΛp∞.
The proof of Theorem 2.3 is complete.


## 3.p-Carleson Measure on Siegel Upper Half Space

LetI be a cube in ℍn with center η. The Carleson box based on I is defined by
(3.1)S(I)={(ξ,ρ)∈𝕌n:|η-1ξ|∞≤l(I)2,ρ≤l(I)}.Forp>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
(3.2)μ(S(I))≤C|I|p,∀cubesI⊂ℍn.Forz∈ℍn, let Γ(z)={(ξ,ρ)∈𝕌n:|z-1ξ|<ρ} be the cone at z. Suppose that f is a measurable function on 𝕌n. The nontangential maximal function N(f) of f is defined by
(3.3)N(f)(z)=sup(ξ,ρ)∈Γ(z)|f(ξ,ρ)|.Since the nontangential maximal function and Hausdorff capacity are defined, we are paying attention on the characterizations of thep-Carleson measures.Theorem 3.1.
Forp∈(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 constantC>0 such that
(3.4)∫𝕌n‍|f(ξ,ρ)|dμ≤C∫ℍn‍N(f)dΛp(∞)holds for all Borel measurable functionsf on 𝕌n.(iii)
For everyq>0, there exists a constant C>0 such that
(3.5)∫𝕌n‍|f(ξ,ρ)|qdμ≤C∫ℍn‍(N(f))qdΛp(∞)
holds for all Borel measurable functionsf 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:N(f)(z)>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 ⋃kJk=⋃jIj, ∑k‍|Jk|p≤∑j‍|Ij|p and T(Eλ)⊂⋃kT(Jk*), where Jk* is the cube with the same center and 5cn times sidelength of Jk.
If(ξ,ρ)∈𝕌n satisfies |f(ξ,ρ)|>λ, then N(f)(z)>λ for all z∈B(ξ,ρ). Thus (ξ,ρ)∈T(Eλ), and hence
(3.6){(ξ,ρ)∈𝕌n:|f(ξ,ρ)|>λ}⊂T(Eλ)⊂⋃k‍T(Jk*).
By Lemma 2.4, we obtain
(3.7)μ({(ξ,ρ)∈𝕌n:|f(ξ,ρ)|>λ})≤μ(⋃k‍T(Jk*))≤∑k‍μ(T(Jk*))≤C∥μ∥∑k‍|Jk*|p≤C∥μ∥∑j‍|Jj*|p.
Taking an infimum over all such dyadic cube coverings, we have
(3.8)μ({(ξ,ρ)∈𝕌n:|f(ξ,ρ)|>λ})≤C∥μ∥Λp(∞)(Eλ).
Therefore,
(3.9)∫𝕌n‍|f(ξ,ρ)|dμ=∫0∞‍μ({(ξ,ρ)∈𝕌n:|f(ξ,ρ)|>λ})dλ≤C∫0∞‍∥μ∥Λp(∞)(Eλ)dλ=C∥μ∥∫ℍn‍N(f)dΛp(∞).
(ii)⇒(i). Suppose that (3.4) is valid for all Borel measurable functions on 𝕌n. For a cube I⊂ℍn, let ϕ(ξ,ρ)=χT(I). Note that (ξ,ρ)∈T(I)∩Γ(z) if and only if z∈B(ξ,ρ)⊂I, there is N(ϕ)=χI. Then, by (3.4), we have
(3.10)μ(T(I))≤C∫0∞‍Λp(∞)({N(ϕ)>λ})dλ=C∫01‍Λp(∞)(I)dλ≤C|I|p.
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
(3.11)∫𝕌n‍|f(ξ,ρ)|qdμ≤C∫ℍn‍(N(f))qdΛp(∞).
(iii)⇒(ii). If we set q=1 in (3.5), then (3.4) holds.
The proof of Theorem3.1 is complete.To continue the characterization of thep-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.Lemma 3.2.
Letf be a locally integrable function on ℍn, and ϕ∈𝒮(ℍn) be a nonnegative radial and decreasing function with ∫ℍn‍ϕ(z)dz=1. Then
(3.12)N(F)(z)=sup(ξ,ρ)∈Γ(z)|F(ξ,ρ)|=sup(ξ,ρ)∈Γ(z)|f*ϕρ(ξ)|≤CM(f)(z),
where ϕρ(ξ)=ρ-(2n+2)ϕ(ρ-1ξ) and ρ>0 is real.Proof.
Sinceϕ∈𝒮(ℍn), we can suppose that |ϕ(z)|≤C/(1+|z|)2n+2+ε for some ε>0. Then
(3.13)|f*ϕρ(ξ)|=|∫ℍn‍ρ-(2n+2)ϕ(ρ-1(η-1ξ))f(η)dη|≤∫|η-1ξ|≤ρ‍ρ-(2n+2)|ϕ(ρ-1(η-1ξ))||f(η)|dη+∑k=0∞‍∫2kρ<|η-1ξ|≤2k+1ρ‍ρ-(2n+2)|ϕ(ρ-1(η-1ξ))||f(η)|dη≤Cρ2n+2∫|η-1ξ|≤ρ‍|f(η)|dη+C∑k=0∞‍∫2kρ<|η-1ξ|≤2k+1ρ‍ρ-(2n+2)|f(η)|(1+((η-1ξ)/ρ))2n+2+εdη≤CM(f)(ξ)+C∑k=0∞‍1ρ2n+2∫|η-1ξ|≤2k+1ρ‍12k(2n+2+ε)|f(η)|dη≤C′M(f)(ξ)∑k=0∞‍2-(k+1)ε=C′′M(f)(ξ).
Hence
(3.14)N(F)(z)=sup(ξ,ρ)∈Γ(z)|f*ϕρ(ξ)|≤CM(f)(z).
Lemma 3.2 is proved.Theorem 3.3.
For0<p≤1, q>p, let μ be a positive Borel measure on 𝕌n. Then μ is a p-Carleson measure if and only if
(3.15)∫𝕌n‍|G(ξ,ρ)|qdμ≤C∫ℍn‍|g|qdΛp(∞)
holds for all functions G on 𝕌n which can be written as G(ξ,ρ)=g*ϕρ(ξ), where g is a locally integrable function on ℍn, and ϕ is the function in Lemma 3.2.Proof.
Suppose thatμ is a p-Carleson measure. By the conclusions of Theorems 3.1 and 2.3 and Lemma 3.2, there is
(3.16)∫𝕌n‍|G(ξ,ρ)|qdμ≤C∫ℍn‍(N(G))qdΛp(∞)≤C∫ℍn‍(M(g))qdΛp(∞)≤C∫ℍn‍|g|qdΛp(∞).
Conversely, for any cubeI⊂ℍn, if we set g=χI, then for (ξ,ρ)∈T(I), there is
(3.17)G(ξ,ρ)=∫I‍ϕρ(η-1ξ)dη≥∫B(ξ,ρ)‍ϕρ(η-1ξ)dη=∫B(ξ,1)‍ϕ1(η-1ξ)dη=∫B(0,1)‍ϕ(z)dz=c.
Thus, by using the inequality (3.15), we obtain
(3.18)cqμ(T(I))=cq∫T(I)‍dμ≤∫T(I)‍|G(ξ,ρ)|qdμ≤∫𝕌n‍|G(ξ,ρ)|qdμ≤C∫ℍn‍|g|qdΛp(∞)=C∫ℍn‍χIdΛp(∞)=CΛp(∞)(I)≤C|I|p.
This ends the proof of Theorem 3.3.Letℕ(Λp(∞)) be the space of all Borel measurable functions f on 𝕌n satisfying
(3.19)∥f∥ℕ(Λp(∞))=∫ℍn‍N(f)dΛp(∞)<∞.
Then ∥·∥ℕ(Λp(∞)) gives a (quasi) norm and ℕ(Λp(∞)) is complete.The following result says thatℕ(Λp(∞)) is the dual of p-Carleson measure.Theorem 3.4.
Letp∈(0,1]. Then there exists a duality between the space of p-Carleson measures and ℕ(Λp(∞)) in the following sense.(i)
Everyp-Carleson measure μ on 𝕌n defines a bounded linear functional on ℕ(Λp(∞)) via the pairing
(3.20)〈μ,f〉=∫𝕌n‍f(ξ,ρ)dμ.(ii)
Letℕ0(Λp(∞)) be the closure in ℕ(Λp(∞)) 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.Proof.
Assume thatμ is a p-Carleson measure and f∈ℕ(Λp(∞)). By (3.19) and (ii) in Theorem 3.1, we have
(3.21)∫𝕌n‍|f(ξ,ρ)|dμ≤C∫ℍn‍N(f)dΛp(∞)<∞.
Thus, 〈μ,f〉=∫𝕌n‍f(ξ,ρ)dμ is well defined. Hence, every p-Carleson measure μ defines a bounded linear functional on ℕ(Λp(∞)). 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
(3.22)L(f)=∫𝕌n‍f(ξ,ρ)dμ=〈μ,f〉.
If f=χT(I) for any cube I∈ℍn, then
(3.23)μ(T(I))=∫𝕌n‍χT(I)dμ=L(χT(I))≤∥L∥·∥χT(I)∥ℕ(Λp(∞))=∥L∥∫ℍn‍N(χT(I))dΛp(∞)=C∥L∥∫I‍dΛp(∞)=C∥L∥Λp(∞)(I)≤C∥L∥·|I|p.
Hence, μ is a p-Carleson measure with ∥μ∥≤C∥L∥. 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-Meyer-Stein tent space onℝn (cf. [4, 6]). Then the atomic decomposition of the tent spaces and the duality of the tent space are discussed.Definition 4.1.
Letp∈(0,1]. A Lebesgue measurable function f on 𝕌n is said to belong to Tp∞ if
(4.1)∥f∥Tp∞=supB⊂ℍn(1|B|p∫T(B)‍|f(ξ,ρ)|2dξdρρ1+2(n+1)(1-p))1/2<∞,
where B runs over all balls in ℍn, and T(B) is a tent based on B.Definition 4.2.
Letp∈(0,1]. The tent space Tp1 consists of all measurable functions f on 𝕌n for which
(4.2)∥f∥Tp1=infw(∫𝕌n‍|f(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p))1/2<∞,
where the infimum is taken over all nonnegative Borel measurable functions w on 𝕌n satisfying
(4.3)∫ℍn‍N(w)dΛp(∞)≤1,
and w is allowed to vanish only where f vanishes.
We will identifyTp∞ with a dual space of Tp1. In order to do this, we first introduce the Tp1-atom as follows.Definition 4.3.
A functiona 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
(4.4)∫T(B)‍|a(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)≤1|B|p.
For the tent spaceTp1 on 𝕌n and ∥·∥Tp1, we have the triangle inequality with a constant in the following lemma.Lemma 4.4.
Letp∈(0,1]. If ∑j‍∥gj∥Tp1<∞, then g=∑j‍gj∈Tp1 and
(4.5)∥g∥Tp1≤C∑j‍∥gj∥Tp1.Proof.
Without loss of generality, we assume thatλj=∥gj∥Tp1>0 for all j. Set fj=∥g∥Tp1-1gj. Then ∥fj∥Tp1≤1 and g=∑j‍λjfj. Suppose that wj≥0 for all j such that ∫ℍn‍N(wj)dΛp(∞)≤1 and
(4.6)∫𝕌n‍|gj(ξ,ρ)|2wj(ξ,ρ)-1dξdρρ1-2(n+1)(1-p)≤2∥gj∥Tp12.
According to the definition of Tp1, the above inequality holds obviously. By using the Cauchy-Schwarz inequality, we have
(4.7)|g|2≤(∑j‍λjwj)(∑j‍λj|fj|2wj-1).
Let w=(∑j‍λj)-1∑j‍λjwj. 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
(4.8)∫ℍn‍N(w)dΛp(∞)≤(∑j‍λj)-1∑j‍λj∫ℍn‍N(wj)dΛp(∞)≤1.
Thus, by the inequality (4.7), we obtain
(4.9)∫𝕌n‍|g(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p)≤C(∑j‍λj)∑j‍λj∫ℍn‍|fj(ξ,ρ)|2wj(ξ,ρ)-1dξdρρ1-2(n+1)(1-p)≤2C(∑j‍λj)∑j‍λj∥fj∥Tp12=C(∑j‍λj)2=C(∑j‍∥gj∥Tp1)2.
Taking the infimum on the left above inequality, we have
(4.10)∥g∥Tp1≤C∑j‍∥gj∥Tp1.
The proof of Lemma 4.4 is complete.Remark 4.5.
By Lemma4.4, 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 spaceTp1 as follows.Theorem 4.6.
Letp∈(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
(4.11)f=∑j‍λjaj.
Moreover,
(4.12)∥f∥Tp1≈inf{∑j‍|λj|:f=∑j‍λjaj},
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 thata is a Tp1-atom on 𝕌n. Then there exists a ball B=B(z,r)⊂ℍn such that supp a⊂T(B) and
(4.13)∫T(B)‍|a(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)≤1|B|p.
Fix ε>0, and let
(4.14)w(ξ,ρ)=hr-(2n+2)pmin{1,(rD(ξ,z))(2n+2)p+ε},
where D(ξ,z) denotes the distance between (ξ,ρ) and (z,0) in 𝕌n, and h is a suitable constant which will be chosen later. Obviously, w(ξ,ρ) is identically equal to hr-(2n+2)p on the upper half ball of radius r (center is (z,0), and D(ξ,z)≤r) and decays radially outside the ball (outside the ball D(ξ,z)>r, and D(ξ,z)→+∞). For ξ∈ℍn, the distance in 𝕌n from the cone Γ(ξ) to (z,0) is c|z-1ξ|=cd(z,ξ). Then
(4.15)N(w)(z)≤hr-(2n+2)pmin{1,(rcd(ξ,z))(2n+2)p+ε}.
Therefore,
(4.16)h-1∫ℍn‍N(w)dΛp(∞)=∫0∞‍Λp(∞)({ξ∈ℍn:h-1N(w)>λ})dλ≤∫0r-(2n+2)p‍Λp(∞)[B(z,1c(rελ)1/(2n+2)p+ε)]dλ≤C∫0r-(2n+2)p‍Λ~p(∞)[B(z,1c(rελ)1/(2n+2)p+ε)]dλ≤C(rε/(2n+2)p+ε)(2n+2)p∫0r-(2n+2)p‍λ-(2n+2)p/((2n+2)p+ε)dλ=C.
Thus, ∫ℍn‍N(w)dΛp(∞)≤1 by choosing h=C-1. On the other hand, let w-1=r(2n+2)p on T(B). We have
(4.17)∫T(B)‍|a(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p)=r(2n+2)p∫T(B)‍|a(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)≤r(2n+2)p|B|-p=C.
Therefore, a∈Tp1 and ∥a∥Tp1≤C.
Now, taking a sum∑j‍λjaj, 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
(4.18)∥f∥Tp1≤∑j‍|λj|∥aj∥Tp1≤C∑j‍|λj|<∞.
That means f=∑j‍λjaj∈Tp1.
Conversely, letf∈Tp1. We choose a Borel measurable function w≥0 on 𝕌n such that (4.3) holds and
(4.19)∫𝕌n‍|f(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p)≤2∥f∥Tp12.
For eachk∈ℤ, let Ek={Z∈ℍn:N(w)(z)>2k}. By Lemma 2.4 and the definition of Λp(∞), it follows that there exists a disjoint dyadic cubes sequence {Ik,j} such that
(4.20)∑j‍|Ik,j|p≤2Λp(∞)(Ek),T(Ek)⊂⋃j‍S*(Ik,j),
where S*(Ik,j)={(ξ,ρ)∈𝕌n:ξ∈Ik,j,ρ<2diam(Ik,j)}. Define
(4.21)Tk,j=S*(Ik,j)∖⋃m>k‍⋃l‍S*(Im,l).
Then Tk,j is disjoint in 𝕌n for different j,k. Therefore,
(4.22)⋃k=-KK‍⋃j‍Tk,j=⋃j‍S*(I-K,j)∖⋃m>k‍⋃l‍S*(Im,l)⊃T(E-K)∖⋃m>k‍⋃l‍S*(Im,l).
Note that ⋃kT(Ek)={(ξ,ρ)∈𝕌n:w(ξ,ρ)>0}, (ξ,ρ)∉T(Ek) implies w(ξ,ρ)≤2k. And each S*(Im,l) is contained in a cube of sidelength 4diam(Im,l) in 𝕌n. By using the subadditivity of the p-Hausdorff capacity and (4.3), there is
(4.23)Λp(∞)(⋃m>k‍⋃l‍S*(Im,l))≤C∑m>k‍∑l‍|Im,l|p≤C∑m>k‍Λp(∞)(Em)→0
as k→∞. Thus, by (4.21),
(4.24)⋃k‍⋃j‍Tk,j⊃⋃k‍T(Ek)∖⋂k‍⋃m>k‍⋃l‍S*(Im,l)={(ξ,ρ)∈𝕌n:w(ξ,ρ)>0}∖E∞,
where E∞ is a set of zero p-Hausdorff capacity, that is,
(4.25)Λp(∞)(E∞)=Λp(∞)(⋂k‍⋃m>k‍⋃l‍S*(Im,l))=0.
Since w is allowed to vanish only where f vanishes, we have f=∑k,j‍f·χTk,j a.e. on 𝕌n. Let
(4.26)ak,j=(|Ik,j*|p∫Tk,j‍|f(ξ,ρ)|2dξdρρ1-2(n+1)(1-p))-1/2f·χTk,j,λk,j=(|Ik,j*|p∫Tk,j‍|f(ξ,ρ)|2dξdρρ1-2(n+1)(1-p))1/2,
where the Ik,j* is a cube as Lemma 2.4, that is, Ik,j*=5cnIk,j.
Thus(4.27)f=∑k,j‍λk,jak,j,a.e.on𝕌n.
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 cl(Ik,j*) (c>1 is a constant). Thus, ak,j is supported in T(Bk,j), and
(4.28)∫T(Bk,j)‍|ak,j(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)=(|Ik,j*|p∫Tk,j‍|f(ξ,ρ)|2dξdρρ1-2(n+1)(1-p))-1∫T(Bk,j)‍|f·χTk,j|2dξdρρ1-2(n+1)(1-p)=|Ik,j*|-p=C|Bk,j|-p.
This means that every ak,j is a Tp1-atom. The remaining is to estimate ∑k,j‍|λk,j|.
Notice the fact that(4.29)w≤2k+1onTk,j⊂(⋃l‍S*(Ik+1,l))c⊂(T(Ek+1))c.
By the Cauchy-Schwarz inequality, (4.19) and (4.20), we have
(4.30)∑k,j‍|λk,j|≤∑k,j‍2(k+1)/2|Ik,j*|p/2(∫Tk,j‍|f(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p))1/2≤(∑k,j‍2k+1|Ik,j*|p)1/2(∑k,j‍∫Tk,j‍|f(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p))1/2≤C∥f∥Tp1(∑k‍2k∑j‍|Ik,j|p)1/2≤C∥f∥Tp1(∑k‍2kΛp(∞)(Ek))1/2=C∥f∥Tp1(∫0+∞‍Λp(∞)({ξ∈ℍn:N(w)>λ})dλ))1/2≤C∥f∥Tp1(∫ℍn‍N(w)dΛp(∞))1/2≤C∥f∥Tp1.
If f=∑k‍λkak and every ak is a Tp1-atom, then, by Lemma 4.4, we obtain
(4.31)∥f∥Tp1≤C∑k‍|λk|∥ak∥Tp1≤C∑k‍|λk|.
Thus
(4.32)∥f∥Tp1≈inf{∑j‍|λj|:f=∑j‍λjaj},
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.
Letp∈(0,1]. Then the dual of Tp1 can be identified with Tp∞ under the pairing
(4.33)〈f,g〉=∫𝕌n‍f(ξ,ρ)g(ξ,ρ)dξdρρ.Proof.
We first show that(4.34)∫𝕌n‍|f(ξ,ρ)g(ξ,ρ)|dξdρρ≤C∥f∥Tp1∥g∥Tp∞
holds for all f∈Tp1 and g∈Tp∞.
In fact, assume thatw 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
(4.35)1|B|p∫T(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p)≤C.
It means that |g(ξ,ρ)|2ρ-2(n+1)(1-p)-1dξdρ is a p-Carleson measure, and ∥μ∥≈∥g∥Tp∞2. Hence, by Theorem 3.1, we obtain
(4.36)∫𝕌n‍w(ξ,ρ)|g(ξ,ρ)|2ρ-2(n+1)(1-p)-1dξdρ≤C∥g∥Tp∞2∫ℍn‍N(w)dΛp(∞)≤C∥g∥Tp∞2.
Thus, for f∈Tp1, by using the Cauchy-Schwarz inequality, there is
(4.37)∫𝕌n‍|f(ξ,ρ)g(ξ,ρ)|dξdρρ≤(∫𝕌n‍|f|2w-1dξdρρ1-2(n+1)(1-p))1/2(∫𝕌n‍|g|2wdξdρρ1+2(n+1)(1-p))1/2≤C(∫𝕌n‍|f(ξ,ρ)|2w(ξ,ρ)-1dξdρρ1-2(n+1)(1-p))1/2·∥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.
LetL be a bounded linear functional on Tp1 and fix a ball B=B(z,r)⊂ℍn. If f is supported in T(B) with f∈L2(T(B),ρ-1dξdρ), then
(4.38)∫T(B)‍|f(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)≤r2(n+1)(1-p)∫T(B)‍|f(ξ,ρ)|2dξdρρ=Cr2(n+1)|B|-p∥f∥L2(T(B),ρ-1dξdρ)2.
Therefore, f is a multiple of a Tp1-atom with
(4.39)∥f∥Tp12≤Cr2(n+1)∥f∥L2(T(B),ρ-1dξdρ)2.
Hence, L induces a bounded linear functional on L2(T(B),ρ-1dξdρ). Thus, there exists a function g which is locally in L2(𝕌n,ρ-1dξdρ) such that
(4.40)L(f)=∫𝕌n‍f(ξ,ρ)g(ξ,ρ)dξdρρ,
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 ∥g∥Tp∞≤C∥L∥.
Now, again fix a ballB⊂ℍn and for every ε>0, let
(4.41)fε(ξ,ρ)=ρ-2(n+1)(1-p)g(ξ,ρ)¯χTε(B)(ξ,ρ),
where Tε(B)=T(B)∩{(ξ,ρ):ρ>ε} is the truncated of T(B).
Sinceg∈L2(T(B)), there is
(4.42)∫T(B)‍|fε(ξ,ρ)|2dξdρρ1-2(n+1)(1-p)=∫Tε(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p)<∞.
Hence, fε is a multiple of a Tp1-atom with
(4.43)∥fε∥Tp12≤Cr2(n+1)p∫Tε(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p),
where C is independent of ε. By (4.40), we obtain
(4.44)∫Tε(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p)=L(fε)≤∥L∥·∥fε∥Tp1≤C∥L∥(|B|p∫Tε(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p))1/2.
Thus
(4.45)(|B|-p∫Tε(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p))1/2≤C∥L∥.
It follows that
(4.46)(|B|-p∫T(B)‍|g(ξ,ρ)|2dξdρρ1+2(n+1)(1-p))1/2≤C∥L∥,
since (4.40) is true for all ε>0. Therefore, g∈Tp∞ and ∥g∥Tp∞≤C∥L∥. 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.



---

*Source: 583156-2012-11-27.xml*