Carleson Measure and Tent Spaces on the Siegel Upper Half Space

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.

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 ℝ^𝑛 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 ℝ^𝑛 (cf. [4, 5]). By using the Choquet integrals with respect to Hausdorff capacity on ℝ^𝑛, 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 𝐿^𝑞_Λ∞𝑝(ℍ^𝑛) 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 ℤ^𝑛, ℝ^𝑛, and ℂ^𝑛 be 𝑛-dimensional ℤ, ℝ, and ℂ, respectively. The Siegel upper half space 𝕌^𝑛 in ℂ^𝑛+1 is defined by 𝕌^𝑛=𝑧=𝑧^′,𝑧_𝑛+1∈ℂ^𝑛+1∶Im𝑧_𝑛+1>𝑧^′2,(1.1) where 𝑧^′=(𝑧₁,𝑧₂,…,𝑧_𝑛)∈𝐶^𝑛,|𝑧^′|²=∑^𝑛_𝑘=1|𝑧_𝑘|². The boundary of 𝕌^𝑛 is 𝜕𝕌^𝑛=𝑧=𝑧^′,𝑧_𝑛+1∈ℂ^𝑛+1∶Im𝑧_𝑛+1=𝑧^′2.(1.2)

The Heisenberg group on ℂ^𝑛×ℝ, denoted by ℍ^𝑛, is a noncommutative nilpotent Lie group with the underlying manifold ℂ^𝑛×ℝ. The group law is given by 𝑧𝑤=𝑧^′,𝑡𝑤^′,𝑠=𝑧^′+𝑤^′,𝑡+𝑠+2Im𝑧^′𝑤^′,for𝑧=𝑧^′,𝑡,𝑤=𝑤^′,𝑠∈ℍ^𝑛,(1.3) where 𝑧^′𝑤^′=𝑧₁𝑤₁+⋯+𝑧_𝑛𝑤_𝑛is the Hermitean product on ℂ^𝑛.

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 ℍ^𝑛 coincides with the Lebesgue measure 𝑑𝑧^′𝑑𝑡 on ℂ^𝑛×ℝ.

For each element 𝜁=(𝜁^′,𝑡) of ℍ^𝑛, the following affine self-mapping of 𝕌^𝑛: 𝜁^′,𝑡∶𝑧^′,𝑧_𝑛+1↦𝑧^′+𝜁^′,𝑧_𝑛+1+𝑡+2𝑖𝑧^′𝜁^′+𝑖𝜁^′2(1.4) is an action of the group ℍ^𝑛 on the space 𝕌^𝑛. Observe that the mapping (1.4) gives us a realization of ℍ^𝑛 as a group of affine holomorphic bijections of 𝕌^𝑛 (see [5]).

The dilations on ℍ^𝑛 are defined by 𝛿𝑧=𝛿(𝑧^′,𝑡)=(𝛿𝑧^′,𝛿²𝑡), 𝛿>0, and the rotation on ℍ^𝑛 is defined by 𝜎𝑧=𝜎(𝑧^′,𝑡)=(𝜎𝑧^′,𝑡) with a unitary map 𝜎 of ℂ^𝑛. The conjugation of 𝑧 is 𝑧=(𝑧^′,𝑡)=(𝑧^′,−𝑡). The norm function is given by |𝑧|=𝑧^′4+|𝑡|^21/4,𝑧=𝑧^′,𝑡∈ℍ^𝑛,(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 𝑑(𝑧,𝑤) of point 𝑧 and 𝑤 in ℍ^𝑛 is defined by 𝑑(𝑧,𝑤)=|𝑤⁻¹𝑧|.

For 𝑧=(𝑧^′,𝑡)∈ℍ^𝑛 (or 𝑧=(𝑥,𝑦,𝑡)∈ℍ^𝑛 for 𝑥=(𝑥₁,𝑥₂,,…,𝑥_𝑛), 𝑦=(𝑦₁,𝑦₂,,…,𝑦_𝑛)), we define |𝑧|_∞=max𝑥₁,…,𝑥_𝑛,𝑦₁,…,𝑦_𝑛,|𝑡|,(1.6) where 𝑧^′=(𝑧₁,…,𝑧_𝑛), and 𝑧_𝑘=𝑥_𝑘+𝑖𝑦_𝑘,1≤𝑘≤𝑛.

To define the cube in ℍ^𝑛, let [𝑥]=([𝑥₁],[𝑥₂],…,[𝑥_𝑛]), where [𝑎] denotes the largest integer less than or equal to 𝑎, and ⟨𝑥⟩=𝑥−[𝑥]. We define the function 𝐹(𝑥,𝑦) by 𝐹𝑥,𝑦=^∞_𝑗=124^𝑗2^𝑗𝑥2^𝑗𝑦mod2ℤ^𝑛−2^𝑗𝑦2^𝑗𝑥mod2ℤ^𝑛,(1.7) and set 𝐼₀=𝑥,𝑦,𝑡∈ℍ^𝑛∶0≤𝑥_𝑘≤1,0≤𝑦_𝑘≤1,1≤𝑘≤𝑛,0≤𝑡−𝐹𝑥,𝑦≤1,(1.8)Γ={(𝑚,𝑛,𝑙)∈ℍ^𝑛∶𝑚,𝑛∈ℤ^𝑛;𝑙∈ℤ}. Then ℍ^𝑛=_𝛾∈Γ𝛾⁻¹𝐼^′₀,disjointunion,(1.9) where 𝐼^′₀={(𝑥,𝑦,𝑡)∈ℍ^𝑛∶0≤𝑥_𝑘<1,0≤𝑦_𝑘<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 𝑐_𝑛𝑙(𝐼), where 𝑐_𝑛 is a constant depending only on 𝑛. The “dyadic cubes" on ℍ^𝑛 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 in ℍ^𝑛 with center 𝑤 and radius 𝑟 is denoted as 𝐵=𝐵(𝑤,𝑟)={𝑧∶|𝑤⁻¹𝑧|<𝑟}. The tent based on the set 𝐸⊂ℍ^𝑛 is defined by 𝑇(𝐸)=𝜉,𝜌∈𝕌^𝑛∶𝐵𝜉,𝜌⊂𝐸.(1.10)

The Schwartz class of rapidly decreasing smooth function on ℍ^𝑛 will be denoted by 𝒮(ℍ^𝑛). The dual of 𝒮(ℍ^𝑛) is 𝒮^′(ℍ^𝑛), the space of tempered distributions on ℍ^𝑛.

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⊂ℍ^𝑛, the 𝑝-dimensional Hausdorff capacity of 𝐸 is defined by Λ^∞_𝑝(𝐸)=inf_𝑗𝐼_𝑗^𝑝∶𝐸⊂^∞_𝑗=1𝐼_𝑗,(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 𝐸_𝑗 is nondecreasing, then Λ^∞_𝑝(⋃_𝑗𝐸_𝑗)=lim_𝑗→∞Λ^∞_𝑝(𝐸_𝑗).(ii)If 𝐸_𝑗 is nonincreasing, then Λ^∞_𝑝(⋂_𝑗𝐸_𝑗)=lim_𝑗→∞Λ^∞_𝑝(𝐸_𝑗).(iii)Λ^∞_𝑝 has the strong subadditivity condition. Λ^∞_𝑝𝐸₁𝐸₂+Λ^∞_𝑝𝐸₁∩𝐸₂≤Λ^∞_𝑝𝐸₁+Λ^∞_𝑝𝐸₂.(2.2)

For a nonnegative function 𝑓 on ℍ^𝑛, the Choquet integral with respect to the Hausdorff capacity on the Heisenberg group is defined by _ℍ𝑛𝑓(𝑧)𝑑Λ^∞_𝑝=^∞₀Λ^∞_𝑝𝑧∈ℍ^𝑛∶𝑓(𝑧)>𝜆𝑑𝜆.(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∈𝐿¹_loc(ℍ^𝑛). 𝑀(𝑓), the dyadic Hardy-Littlewood maximal operator of 𝑓, is defined by 𝑀𝑓(𝑧)=sup_𝐼∋𝑧1|𝐼|_𝐼𝑓(𝜉)𝑑𝜉,(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 integral spaces 𝐿^𝑞_Λ∞𝑝(ℍ^𝑛) is as follows.

Theorem 2.3. Suppose 0<𝑝≤1. There exists a constant 𝐶 depending only on 𝑝 and 𝑛 such that _ℍ𝑛𝑀𝑓^𝑞𝑑Λ^∞_𝑝≤𝐶_ℍ𝑛𝑓^𝑞𝑑Λ^∞_𝑝,∀𝑓∈𝐿^𝑞_Λ∞𝑝ℍ^𝑛Λ,𝑞>𝑝,(2.5)_∞𝑝𝜉∈ℍ^𝑛𝑓(𝜉)≤𝐶∶𝑀>𝜆𝜆^𝑝_ℍ^𝑛𝑓^𝑝𝑑Λ^∞_𝑝,∀𝑓∈𝐿^𝑝_Λ∞𝑝ℍ^𝑛.(2.6)

To prove Theorem 2.3, we need to prove the following lemmas first.

Lemma 2.4. For 𝑝∈(0,1], let {𝐼_𝑗} be a sequence of dyadic cubes in ℍ^𝑛 such that ∑_𝑗|𝐼_𝑗|^𝑝<∞. Then there is a sequence of disjoint dyadic cubes {𝐽_𝑘} so that ⋃_𝑘𝐽_𝑘=⋃_𝑗𝐼_𝑗 and ∑_𝑘|𝐽_𝑘|^𝑝≤∑_𝑗|𝐼_𝑗|^𝑝. Moreover, if 𝐸⊂⋃_𝑗𝐼_𝑗, then the following tent inclusion 𝑇(𝐸)⊂⋃_𝑘𝑇(𝐽^∗_𝑘) holds, where 𝐽^∗_𝑘 is the cube with the same center as 𝐽_𝑘 and 𝛿_𝑛 times the sidelength (𝛿_𝑛 is some constant depending only on 𝑛).

Proof. Since ∑_𝑗|𝐼_𝑗|^𝑝<∞, 𝑝∈(0,1], we have that ∑_𝑗|𝐼_𝑗|<∞. Thus the union of 𝐼_𝑗 cannot form arbitrarily large dyadic cubes. Thus, each 𝐼_𝑗 must be included in some maximal dyadic cube 𝐽_𝑘∈𝕁, where 𝕁 denotes the collection of all dyadic cubes 𝐽=⋃{𝐼_𝑗∶𝐼_𝑗⊂𝐽}. If we write these maximal cubes as a sequence {𝐽_𝑘}, then 𝐽_𝑘 is disjoint and ⋃_𝑘𝐽_𝑘=⋃_𝑗𝐼_𝑗. By the definition of 𝐽_𝑘 in 𝕁, we know that |𝐽_𝑘|≤∑_{𝐼𝑗⊂𝐽𝑘}|𝐼_𝑗|. Jensen’s inequality gives |𝐽_𝑘|^𝑝≤∑_{𝐼𝑗⊂𝐽𝑘}|𝐼_𝑗|^𝑝 for 0<𝑝≤1. Consequently, _𝑘𝐽_𝑘^𝑝≤_{𝑘𝐼𝑗⊂𝐽𝑘}𝐼_𝑗^𝑝≤_𝑗𝐼_𝑗^𝑝.(2.7)
Suppose that 𝐸⊂⋃_𝑗𝐼_𝑗 and (𝜉,𝜌)∈𝑇(𝐸). Then 𝜉∈𝐸⊂⋃_𝑘𝐽_𝑘 and 𝜉∈𝐽_𝑘 for some fixed 𝑘. For this 𝐽_𝑘, we consider the parent dyadic cube 𝐽^′_𝑘, namely, the unique dyadic cube containing 𝐽_𝑘 whose sidelength is double of that of 𝐽_𝑘. Since 𝐽_𝑘 is maximal in 𝕁, we have that 𝐽^′_𝑘 is not a union of the 𝐼_𝑗’s, that is, 𝐽^′_𝑘 contains a point 𝜂∈ℍ^𝑛⧵(⋃_𝑗𝐼_𝑗)⊂ℍ^𝑛⧵𝐸. Denoting the boundary of 𝐸 by 𝜕𝐸, there is 𝜌<dist(𝜉,𝜕𝐸)≤diam𝐽^′_𝑘=𝑐_𝑛𝑙𝐽^′_𝑘=2𝑐_𝑛𝑙𝐽_𝑘.(2.8) If 𝐽^∗_𝑘 is the cube with the same center as 𝐽_𝑘, and 5𝑐_𝑛 times the sidelength, then dist𝜉,𝜕𝐽^∗_𝑘≥125𝑐_𝑛−1𝑙𝐽_𝑘≥2𝑐_𝑛𝑙𝐽_𝑘>𝜌,since𝑐_𝑛>1,(2.9) which means that (𝜉,𝜌)∈𝑇(𝐽^∗_𝑘). The proof is completed.

Lemma 2.5. Let 𝜒_𝐼 be the characteristic function on the cube 𝐼. Then _ℍ𝑛𝑀𝜒_𝐼^𝑞𝑑Λ^∞_𝑝≤𝐶|𝐼|^𝑝,𝑞>𝑝.(2.10)

Proof. Let 𝑧_𝐼 be the center of 𝐼. By the definition of the maximal function 𝑀, we obtain 𝑀𝜒_𝐼(𝜉)|𝐼|≤𝐶inf1,𝑧_𝐼⁻¹𝜉^(2𝑛+2).(2.11) Since 𝑝/𝑞<1, there is _ℍ𝑛𝑀𝜒_𝐼^𝑞𝑑Λ^∞_𝑝≤𝐶|𝐼|^𝑝+𝐶¹₀|𝐼|^𝑝𝜆^−𝑝/𝑞𝑑𝜆=𝐶|𝐼|^𝑝.(2.12) The proof of Lemma 2.5 is complete.

Lemma 2.6. Let {𝐼_𝑗} be a family of nonoverlapping dyadic cubes. Then there is a maximal subfamily {𝐼_𝑗𝑘} such that for every dyadic cube 𝐼, _{𝐼𝑗𝑘⊂𝐼}𝐼_𝑗𝑘^𝑝≤2|𝐼|^𝑝,Λ(2.13)_∞𝑝𝐼_𝑗≤2_𝑘𝐼_𝑗𝑘^𝑝.(2.14)

Proof. Similar to the proof in [16], if 𝐼_𝑗1=𝐼₁, then obviously 𝐼_𝑗1 satisfies (2.13). If 𝑗₁,…,𝑗_𝑘 have been chosen so that (2.13) holds, then we define 𝑗_𝑘+1 as the first index such that the family {𝐼_𝑗1,…,𝐼_𝑗𝑘,𝐼_𝑗𝑘+1} satisfies (2.13). Continuing this proceeding, therefore, we have that {𝐼_𝑗𝑘} is a maximal subfamily of {𝐼_𝑗} satisfying (2.13). Hence (2.13) holds.
To prove (2.14), let 𝑗 be an index such that 𝑗_𝑚<𝑗<𝑗_𝑚+1 for some 𝑚∈ℤ. Then by the proof of (2.13), there exists a dyadic cube 𝐼^∗_𝑗⊃𝐼_𝑗 such that _{𝐼𝑗𝑘⊂𝐼∗𝑗,𝑘≤𝑚}𝐼_𝑗𝑘^𝑝+𝐼_𝑗^𝑝>2𝐼^∗_𝑗^𝑝.(2.15) Therefore, 𝐼^∗_𝑗^𝑝≤_{𝐼𝑗𝑘⊂𝐼∗𝑗,𝑘≤𝑚}𝐼_𝑗𝑘^𝑝.(2.16) We can assume that ∑_𝑘|𝐼_𝑗𝑘|^𝑝<∞. Otherwise (2.14) is obviously correct. Then the sequence {|𝐼^∗_𝑗|} is bounded. Thus, we can consider the family {𝐼_𝑙̃} of maximal cubes of the family {𝐼^∗_𝑗}. Hence _𝑗𝐼_𝑗⊂_𝑘𝐼_𝑗𝑘𝑙𝐼_𝑙̃.(2.17) Consequently, by the definition of Λ^∞_𝑝, we obtain Λ^∞_𝑝𝐼_𝑗≤2_𝑘𝐼_𝑗𝑘^𝑝.(2.18) The proof is complete.

Proof of Theorem 2.3. By the definition of Λ^∞_𝑝, for any 𝜀>0, there exist {𝐼_𝑗}_𝑗 such that _𝑗𝐼_𝑗^𝑝<Λ^∞_𝑝𝐸_𝑘+𝜀,(2.19) where 𝐸_𝑘={𝜉∈ℍ^𝑛∶2^𝑘<|𝑓(𝜉)|≤2^𝑘+1}. If Λ^∞_𝑝(𝐸_𝑘)=0, we can choose ∑_𝑗|𝐼_𝑗|^𝑝=0. If Λ^∞_𝑝(𝐸_𝑘)>0, then _𝑗𝐼_𝑗^𝑝≤2Λ^∞_𝑝𝐸_𝑘.(2.20) By Lemma 2.4, for each integer 𝑘, there is a family of nonoverlapping dyadic cubes {𝐼_𝑗^(𝑘)} such that 𝜉∈ℍ^𝑛∶2^𝑘<𝑓(𝜉)≤2^𝑘+1⊂_𝑗𝐼_𝑗^(𝑘),_𝑗𝐼_𝑗^(𝑘)𝑝≤2Λ^∞_𝑝𝜉∈ℍ^𝑛∶2^𝑘<𝑓(𝜉)≤2^𝑘+1.(2.21) Set 𝑔=∑_𝑘2^(𝑘+1)𝑞𝜒_𝐹𝑘, where 𝜒_𝐹𝑘 is the characteristic function of 𝐹_𝑘=⋃_𝑗𝐼_𝑗^(𝑘). Then |𝑓|^𝑞≤𝑔.
First, assume that 𝑞>1. Then the Hölder inequality tells us 𝑀𝑓^𝑞≤𝑀𝑓^𝑞≤𝑀𝑔≤_𝑘2^(𝑘+1)𝑞_𝑗𝑀𝜒_𝐼𝑗^(𝑘).(2.22) Therefore, by Lemma 2.5, we have _ℍ𝑛𝑀𝑓^𝑞𝑑Λ^∞_𝑝≤𝐶_𝑘2^(𝑘+1)𝑞_𝑗ℍ𝑛𝑀𝜒_𝐼𝑗^(𝑘)𝑑Λ^∞_𝑝≤𝐶_𝑘2^(𝑘+1)𝑞_𝑗𝐼_𝑗^(𝑘)𝑝≤𝐶_𝑘2^(𝑘+1)𝑞Λ^∞_𝑝𝐸_𝑘≤𝐶_𝑘2^2𝑞2^𝑞−1^2𝑘𝑞₂^{(𝑘−1)𝑞}Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑞>𝜆𝑑𝜆≤𝐶^∞₀Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑞>𝜆𝑑𝜆=𝐶_ℍ𝑛𝑓^𝑞𝑑Λ^∞_𝑝.(2.23) Now, assume that 𝑝<𝑞≤1. Since |𝑓|≤∑_𝑘2^𝑘+1𝜒_𝐹𝑘, there is 𝑀𝑓≤_𝑘2^𝑘+1_𝑗𝑀𝜒_𝐼𝑗^(𝑘).(2.24) By using Jensen’s inequality, we obtain 𝑀𝑓^𝑞≤_𝑘2^(𝑘+1)𝑞_𝑗𝑀𝜒_𝐼𝑗^(𝑘)𝑞.(2.25) Thus, also by Lemma 2.5, there is _ℍ𝑛𝑀𝑓^𝑞𝑑Λ^∞_𝑝≤𝐶_𝑘2^(𝑘+1)𝑞_𝑗ℍ𝑛𝑀𝜒_𝐼𝑗^(𝑘)𝑑Λ^∞_𝑝≤𝐶_𝑘2^(𝑘+1)𝑞_𝑗𝐼_𝑗^(𝑘)𝑝≤𝐶_𝑘2^(𝑘+1)𝑞Λ^∞_𝑝𝐸_𝑘≤𝐶^∞₀Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑞>𝜆𝑑𝜆=𝐶_ℍ𝑛𝑓^𝑞𝑑Λ^∞_𝑝.(2.26) That means that (2.5) holds.
For a given 𝜆>0, let {𝐼_𝑗} be the family of maximal dyadic cubes 𝐼_𝑗 such that 1𝐼_𝑗^𝐼_𝑗𝑓(𝜉)𝑑𝜉>𝜆.(2.27) Note that 𝑀(𝑓) is dyadic and {𝑧∈ℍ^𝑛∶𝑀(𝑓)(𝑧)>𝜆}=⋃_𝑗𝐼_𝑗. Since 0<𝑝≤1, there is _𝑗𝐼_𝑗≤_𝑗𝐼_𝑗^𝑝1/𝑝,Λ^∞₁(𝐸)≤Λ^∞_𝑝(𝐸)^1/𝑝.(2.28) Hence _ℍ𝑛𝑓(𝑧)𝑑𝑧=^∞₀Λ^∞₁𝜉∈ℍ^𝑛∶𝑓(𝜉=1)>𝜆𝑑𝜆𝑝^∞₀Λ^∞₁𝜉∈ℍ^𝑛∶𝑓(𝜉)>𝜌^1/𝑝𝜌^(1/𝑝)−1≤1𝑑𝜌𝑝^∞₀Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)>𝜌^1/𝑝1/𝑝𝜌^(1/𝑝)−1=1𝑑𝜌𝑝^∞₀Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)|^𝑝>𝜌𝜌^(1/𝑝)−1Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑝≤1>𝜌𝑑𝜌𝑝_ℍ^𝑛𝑓^𝑝𝑑Λ^∞_𝑝^{(1/𝑝)−1∞}₀Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑝=1>𝜌𝑑𝜌𝑝_ℍ^𝑛𝑓^𝑝𝑑Λ^∞_𝑝^1/𝑝,(2.29) where the last inequality is due to the fact that Λ^∞_𝑝𝜉∈ℍ^𝑛∶𝑓(𝜉)^𝑝>𝜌𝜌≤_{{𝜉∈ℍ𝑛∶|𝑓(𝜉)|𝑝>𝜌}}𝑓^𝑝𝑑Λ^∞_𝑝,𝜌>0.(2.30) Therefore, by (2.27) and (2.29), we obtain 𝐼_𝑗^𝑝≤1𝜆_𝐼𝑗𝑓𝑑𝜉^𝑝≤𝐶𝜆^−𝑝_𝐼𝑗𝑓^𝑝𝑑Λ^∞_𝑝.(2.31) For the above {𝐼_𝑗}, by Lemma 2.6 and (2.31), there exists a subfamily {𝐼_𝑗𝑘} such that Λ^∞_𝑝𝑧∈ℍ^𝑛∶𝑀𝑓(𝑧)>𝜆=Λ^∞_𝑝𝑗𝐼_𝑗≤2_𝑘𝐼_𝑗𝑘^𝑝≤𝐶𝜆^−𝑝_{𝑘𝐼𝑗𝑘}𝑓^𝑝𝑑Λ^∞_𝑝≤𝐶𝜆^−𝑝_ℍ𝑛𝑓^𝑝𝑑Λ^∞_𝑝.(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 ℍ^𝑛 with center 𝜂. The Carleson box based on 𝐼 is defined by 𝑆(𝐼)=𝜉,𝜌∈𝕌^𝑛∶𝜂⁻¹𝜉_∞≤𝑙(𝐼)2,𝜌≤𝑙(𝐼).(3.1)

For 𝑝>0, a positive Borel measure 𝜇 on the Siegel upper half space 𝕌^𝑛 is called a 𝑝-Carleson measure if there exists a constant 𝐶>0 such that 𝜇(𝑆(𝐼))≤𝐶|𝐼|^𝑝,∀cubes𝐼⊂ℍ^𝑛.(3.2)

For 𝑧∈ℍ^𝑛, let Γ(𝑧)={(𝜉,𝜌)∈𝕌^𝑛∶|𝑧⁻¹𝜉|<𝜌} be the cone at 𝑧. Suppose that 𝑓 is a measurable function on 𝕌^𝑛. The nontangential maximal function 𝑁(𝑓) of 𝑓 is defined by 𝑁𝑓(𝑧)=sup_{(𝜉,𝜌)∈Γ(𝑧)}𝑓𝜉,𝜌.(3.3)

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 𝕌^𝑛. Then the following three conclusions are equivalent.(i)𝜇 is a 𝑝-Carleson measure.(ii) There exists a constant 𝐶>0 such that _𝕌𝑛𝑓𝜉,𝜌𝑑𝜇≤𝐶_ℍ𝑛𝑁𝑓𝑑Λ_𝑝^(∞)(3.4) holds for all Borel measurable functions 𝑓 on 𝕌^𝑛.(iii) For every 𝑞>0, there exists a constant 𝐶>0 such that _𝕌𝑛𝑓𝜉,𝜌^𝑞𝑑𝜇≤𝐶_ℍ𝑛𝑁𝑓^𝑞𝑑Λ_𝑝^(∞)(3.5)
holds for all Borel measurable functions 𝑓 on 𝕌^𝑛.

Proof. (i)⇒(ii). Assume that 𝜇 is a 𝑝-Carleson measure, and 𝑓 is a Borel measurable function on 𝕌^𝑛. For 𝜆>0, let 𝐸_𝜆={𝑧∈ℍ^𝑛∶𝑁(𝑓)(𝑧)>𝜆}. If the integral on the right hand side of (3.4) is finite, we may assume that Λ_𝑝^(∞)(𝐸_𝜆)<∞. Let {𝐼_𝑗} be any of the dyadic cubes covering of 𝐸_𝜆 with ∑_𝑗|𝐼_𝑗|^𝑝<∞. Then Lemma 2.4 tells us that there is a sequence {𝐽_𝑘} of dyadic cubes with mutually disjointed so that ⋃_𝑘𝐽_𝑘=⋃_𝑗𝐼_𝑗, ∑_𝑘|𝐽_𝑘|^𝑝≤∑_𝑗|𝐼_𝑗|^𝑝 and 𝑇(𝐸_𝜆)⊂⋃_𝑘𝑇(𝐽^∗_𝑘), where 𝐽^∗_𝑘 is the cube with the same center and 5𝑐_𝑛 times sidelength of 𝐽_𝑘.
If (𝜉,𝜌)∈𝕌^𝑛 satisfies |𝑓(𝜉,𝜌)|>𝜆, then 𝑁(𝑓)(𝑧)>𝜆 for all 𝑧∈𝐵(𝜉,𝜌). Thus (𝜉,𝜌)∈𝑇(𝐸_𝜆), and hence 𝜉,𝜌∈𝕌^𝑛∶𝑓𝜉,𝜌>𝜆⊂𝑇𝐸_𝜆⊂_𝑘𝑇𝐽^∗_𝑘.(3.6) By Lemma 2.4, we obtain 𝜇𝜉,𝜌∈𝕌^𝑛∶𝑓𝜉,𝜌>𝜆≤𝜇_𝑘𝑇𝐽^∗_𝑘≤_𝑘𝜇𝑇𝐽^∗_𝑘≤𝐶𝜇_𝑘𝐽^∗_𝑘^𝑝≤𝐶𝜇_𝑗𝐽^∗_𝑗^𝑝.(3.7) Taking an infimum over all such dyadic cube coverings, we have 𝜇𝜉,𝜌∈𝕌^𝑛∶𝑓𝜉,𝜌>𝜆≤𝐶𝜇Λ_𝑝^(∞)𝐸_𝜆.(3.8) Therefore, _𝕌𝑛𝑓𝜉,𝜌𝑑𝜇=^∞₀𝜇𝜉,𝜌∈𝕌^𝑛∶𝑓𝜉,𝜌>𝜆𝑑𝜆≤𝐶^∞₀𝜇Λ_𝑝^(∞)𝐸_𝜆𝑑𝜆=𝐶𝜇_ℍ𝑛𝑁𝑓𝑑Λ_𝑝^(∞).(3.9)
(ii)⇒(i). Suppose that (3.4) is valid for all Borel measurable functions on 𝕌^𝑛. For a cube 𝐼⊂ℍ^𝑛, let 𝜙(𝜉,𝜌)=𝜒_𝑇(𝐼). Note that (𝜉,𝜌)∈𝑇(𝐼)∩Γ(𝑧) if and only if 𝑧∈𝐵(𝜉,𝜌)⊂𝐼, there is 𝑁(𝜙)=𝜒_𝐼. Then, by (3.4), we have 𝜇(𝑇(𝐼))≤𝐶^∞₀Λ_𝑝^(∞)𝑁𝜙>𝜆𝑑𝜆=𝐶¹₀Λ_𝑝^(∞)(𝐼)𝑑𝜆≤𝐶|𝐼|^𝑝.(3.10) 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 _𝕌𝑛𝑓𝜉,𝜌^𝑞𝑑𝜇≤𝐶_ℍ𝑛𝑁𝑓^𝑞𝑑Λ_𝑝^(∞).(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.

Lemma 3.2. Let 𝑓 be a locally integrable function on ℍ^𝑛, and 𝜙∈𝒮(ℍ^𝑛) be a nonnegative radial and decreasing function with ∫_ℍ𝑛𝜙(𝑧)𝑑𝑧=1. Then 𝑁(𝐹)(𝑧)=sup_{(𝜉,𝜌)∈Γ(𝑧)}𝐹𝜉,𝜌=sup_{(𝜉,𝜌)∈Γ(𝑧)}𝑓∗𝜙_𝜌(𝜉)≤𝐶𝑀𝑓(𝑧),(3.12) where 𝜙_𝜌(𝜉)=𝜌^−(2𝑛+2)𝜙(𝜌⁻¹𝜉) and 𝜌>0 is real.

Proof. Since 𝜙∈𝒮(ℍ^𝑛), we can suppose that |𝜙(𝑧)|≤𝐶/(1+|𝑧|)^2𝑛+2+𝜀 for some 𝜀>0. Then 𝑓∗𝜙_𝜌(𝜉)=_ℍ𝑛𝜌^−(2𝑛+2)𝜙𝜌⁻¹𝜂⁻¹𝜉𝑓𝜂≤𝑑𝜂_{|𝜂−1𝜉|≤𝜌}𝜌^−(2𝑛+2)𝜙𝜌⁻¹𝜂⁻¹𝜉𝑓𝜂+𝑑𝜂_{∞𝑘=02𝑘𝜌<|𝜂−1𝜉|≤2𝑘+1𝜌}𝜌^−(2𝑛+2)𝜙𝜌⁻¹𝜂⁻¹𝜉𝑓𝜂≤𝐶𝑑𝜂𝜌^2𝑛+2_|𝜂⁻¹_𝜉|≤𝜌𝑓𝜂𝑑𝜂+𝐶^∞_{𝑘=02𝑘𝜌<|𝜂−1𝜉|≤2𝑘+1𝜌}𝜌^−(2𝑛+2)𝑓𝜂1+𝜂⁻¹𝜉/𝜌^2𝑛+2+𝜀𝑓(𝜉)𝑑𝜂≤𝐶𝑀+𝐶^∞_𝑘=01𝜌^2𝑛+2_|𝜂⁻¹_𝜉|≤2^𝑘+1_𝜌12^{𝑘(2𝑛+2+𝜀)}𝑓𝜂𝑑𝜂≤𝐶^′𝑀𝑓(𝜉)^∞_𝑘=02^{−(𝑘+1)𝜀}=𝐶^′′𝑀𝑓(𝜉).(3.13) Hence 𝑁(𝐹)(𝑧)=sup_{(𝑧)𝜉,𝜌∈Γ}𝑓∗𝜙_𝜌(𝜉)≤𝐶𝑀𝑓(𝑧).(3.14) Lemma 3.2 is proved.

Theorem 3.3. For 0<𝑝≤1, 𝑞>𝑝, let 𝜇 be a positive Borel measure on 𝕌^𝑛. Then 𝜇 is a 𝑝-Carleson measure if and only if _𝕌𝑛𝐺𝜉,𝜌^𝑞𝑑𝜇≤𝐶_ℍ𝑛𝑔^𝑞𝑑Λ_𝑝^(∞)(3.15) holds for all functions 𝐺 on 𝕌^𝑛 which can be written as 𝐺(𝜉,𝜌)=𝑔∗𝜙_𝜌(𝜉), where 𝑔 is a locally integrable function on ℍ^𝑛, and 𝜙 is the function in Lemma 3.2.

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 _𝕌𝑛𝐺𝜉,𝜌^𝑞𝑑𝜇≤𝐶_ℍ𝑛(𝑁(𝐺))^𝑞𝑑Λ_𝑝^(∞)≤𝐶_ℍ𝑛𝑀𝑔^𝑞𝑑Λ_𝑝^(∞)≤𝐶_ℍ𝑛𝑔^𝑞𝑑Λ_𝑝^(∞).(3.16)
Conversely, for any cube 𝐼⊂ℍ^𝑛, if we set 𝑔=𝜒_𝐼, then for (𝜉,𝜌)∈𝑇(𝐼), there is 𝐺𝜉,𝜌=_𝐼𝜙_𝜌𝜂⁻¹𝜉𝑑𝜂≥_{𝐵(𝜉,𝜌)}𝜙_𝜌𝜂⁻¹𝜉𝑑𝜂=_𝐵(𝜉,1)𝜙₁𝜂⁻¹𝜉𝑑𝜂=_𝐵(0,1)𝜙(𝑧)𝑑𝑧=𝑐.(3.17) Thus, by using the inequality (3.15), we obtain 𝑐^𝑞𝜇(𝑇(𝐼))=𝑐^𝑞_𝑇(𝐼)𝑑𝜇≤_𝑇(𝐼)𝐺𝜉,𝜌^𝑞𝑑𝜇≤_𝕌𝑛𝐺𝜉,𝜌^𝑞𝑑𝜇≤𝐶_ℍ𝑛𝑔^𝑞𝑑Λ_𝑝^(∞)=𝐶_ℍ𝑛𝜒_𝐼𝑑Λ_𝑝^(∞)=𝐶Λ_𝑝^(∞)(𝐼)≤𝐶|𝐼|^𝑝.(3.18) This ends the proof of Theorem 3.3.

Let ℕ(Λ_𝑝^(∞)) be the space of all Borel measurable functions 𝑓 on 𝕌^𝑛 satisfying 𝑓_ℕ(Λ𝑝^(∞)₎=_ℍ𝑛𝑁𝑓𝑑Λ_𝑝^(∞)<∞.(3.19) Then ‖⋅‖_ℕ(Λ𝑝^(∞)₎ gives a (quasi) norm and ℕ(Λ_𝑝^(∞)) is complete.

The following result says that ℕ(Λ_𝑝^(∞)) 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 ℕ(Λ_𝑝^(∞)) in the following sense.(i) Every 𝑝-Carleson measure 𝜇 on 𝕌^𝑛 defines a bounded linear functional on ℕ(Λ_𝑝^(∞)) via the pairing 𝜇,𝑓=_𝕌𝑛𝑓𝜉,𝜌𝑑𝜇.(3.20)(ii) Let ℕ₀(Λ_𝑝^(∞)) be the closure in ℕ(Λ_𝑝^(∞)) of the continuous functions with compact support in 𝕌^𝑛. Then every bounded linear functional on ℕ₀(Λ_𝑝^(∞)) given via the pairing (3.20) by a Borel measure 𝜇 on 𝕌^𝑛 is a 𝑝-Carleson measure.

Proof. Assume that 𝜇 is a 𝑝-Carleson measure and 𝑓∈ℕ(Λ_𝑝^(∞)). By (3.19) and (ii) in Theorem 3.1, we have _𝕌𝑛𝑓𝜉,𝜌𝑑𝜇≤𝐶_ℍ𝑛𝑁𝑓𝑑Λ_𝑝^(∞)<∞.(3.21) Thus, ⟨𝜇,𝑓⟩=∫_𝕌𝑛𝑓(𝜉,𝜌)𝑑𝜇 is well defined. Hence, every 𝑝-Carleson measure 𝜇 defines a bounded linear functional on ℕ(Λ_𝑝^(∞)). The part (i) is proved.
For part (ii), let 𝐿 be a bounded linear functional on ℕ₀(Λ_𝑝^(∞)). By the continuity and the closure of ℕ₀(Λ_𝑝^(∞)), applying the Riesz representation theorem, we obtain a Borel measure 𝜇 on 𝕌^𝑛 having 𝐿𝑓=_𝕌𝑛𝑓𝜉,𝜌𝑑𝜇=𝜇,𝑓.(3.22) If 𝑓=𝜒_𝑇(𝐼) for any cube 𝐼∈ℍ^𝑛, then 𝜇(𝑇(𝐼))=_𝕌𝑛𝜒_𝑇(𝐼)𝑑𝜇=𝐿𝜒_𝑇(𝐼)≤‖𝐿‖⋅𝜒_{𝑇(𝐼)ℕ(Λ𝑝(∞))}=‖𝐿‖_ℍ𝑛𝑁𝜒_𝑇(𝐼)𝑑Λ_𝑝^(∞)=𝐶‖𝐿‖_𝐼𝑑Λ_𝑝^(∞)=𝐶‖𝐿‖Λ_𝑝^(∞)(𝐼)≤𝐶‖𝐿‖⋅|𝐼|^𝑝.(3.23) 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 on ℝ^𝑛 (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 𝕌^𝑛 is said to belong to 𝑇^∞_𝑝 if 𝑓_𝑇∞𝑝=sup_{𝐵⊂ℍ𝑛}1|𝐵|^𝑝_𝑇(𝐵)𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)1/2}<∞,(4.1) where 𝐵 runs over all balls in ℍ^𝑛, and 𝑇(𝐵) is a tent based on 𝐵.

Definition 4.2. Let 𝑝∈(0,1]. The tent space 𝑇¹_𝑝 consists of all measurable functions 𝑓 on 𝕌^𝑛 for which 𝑓_𝑇1𝑝=inf_𝑤𝕌𝑛𝑓𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2}<∞,(4.2) where the infimum is taken over all nonnegative Borel measurable functions 𝑤 on 𝕌^𝑛 satisfying _ℍ𝑛𝑁(𝑤)𝑑Λ_𝑝^(∞)≤1,(4.3) and 𝑤 is allowed to vanish only where 𝑓 vanishes.
We will identify 𝑇^∞_𝑝 with a dual space of 𝑇¹_𝑝. In order to do this, we first introduce the 𝑇¹_𝑝-atom as follows.

Definition 4.3. A function 𝑎 on 𝕌^𝑛 is said to be a 𝑇¹_𝑝-atom, if there exists a ball 𝐵⊂ℍ^𝑛 such that 𝑎 is supported in the tent 𝑇(𝐵) and satisfies _𝑇(𝐵)𝑎𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤1|𝐵|^𝑝.(4.4)
For the tent space 𝑇¹_𝑝 on 𝕌^𝑛 and ‖⋅‖_𝑇¹_𝑝, we have the triangle inequality with a constant in the following lemma.

Lemma 4.4. Let 𝑝∈(0,1]. If ∑_𝑗‖𝑔_𝑗‖_𝑇¹_𝑝<∞, then 𝑔=∑_𝑗𝑔_𝑗∈𝑇¹_𝑝 and 𝑔_𝑇1𝑝≤𝐶_𝑗𝑔_{𝑗𝑇1𝑝}.(4.5)

Proof. Without loss of generality, we assume that 𝜆_𝑗=‖𝑔_𝑗‖_𝑇¹_𝑝>0 for all 𝑗. Set 𝑓_𝑗=‖𝑔‖_𝑇⁻¹_1𝑝𝑔_𝑗. Then ‖𝑓_𝑗‖_𝑇¹_𝑝≤1 and 𝑔=∑_𝑗𝜆_𝑗𝑓_𝑗. Suppose that 𝑤_𝑗≥0 for all 𝑗 such that ∫_ℍ𝑛𝑁(𝑤_𝑗)𝑑Λ_𝑝^(∞)≤1 and _𝕌𝑛𝑔_𝑗𝜉,𝜌²𝑤_𝑗𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤2𝑔_𝑗²_𝑇1𝑝.(4.6) According to the definition of 𝑇¹_𝑝, the above inequality holds obviously. By using the Cauchy-Schwarz inequality, we have 𝑔²≤_𝑗𝜆_𝑗𝑤_𝑗𝑗𝜆_𝑗𝑓_𝑗²𝑤_𝑗⁻¹.(4.7) Let 𝑤=(∑_𝑗𝜆_𝑗)⁻¹∑_𝑗𝜆_𝑗𝑤_𝑗. Notice that the vanishing of 𝑤 implies the vanishing of all 𝑤_𝑗, which can only happen whenever all the 𝑔_𝑗 vanish, that is, 𝑔=0, then _ℍ𝑛𝑁(𝑤)𝑑Λ_𝑝^(∞)≤_𝑗𝜆_𝑗⁻¹_𝑗𝜆_𝑗ℍ𝑛𝑁𝑤_𝑗𝑑Λ_𝑝^(∞)≤1.(4.8) Thus, by the inequality (4.7), we obtain _𝕌𝑛𝑔𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤𝐶_𝑗𝜆_𝑗𝑗𝜆_𝑗ℍ𝑛𝑓_𝑗𝜉,𝜌²𝑤_𝑗𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤2𝐶_𝑗𝜆_𝑗𝑗𝜆_𝑗𝑓_𝑗²_𝑇1𝑝=𝐶_𝑗𝜆_𝑗²=𝐶_𝑗𝑔_{𝑗𝑇1𝑝}².(4.9) Taking the infimum on the left above inequality, we have 𝑔_𝑇1𝑝≤𝐶_𝑗𝑔_{𝑗𝑇1𝑝}.(4.10) The proof of Lemma 4.4 is complete.

Remark 4.5. By Lemma 4.4, one can show that ‖⋅‖_𝑇¹_𝑝 is a quasinorm and the tent space 𝑇¹_𝑝 is complete.
The main result of this section is the atomic decomposition of the tent space 𝑇¹_𝑝 as follows.

Theorem 4.6. Let 𝑝∈(0,1]. Then a function 𝑓 on 𝕌^𝑛 belongs to 𝑇¹_𝑝 if and only if there exist a sequence of 𝑇¹_𝑝-atoms {𝑎_𝑗} and an 𝑙¹-sequence {𝜆_𝑗} such that 𝑓=_𝑗𝜆_𝑗𝑎_𝑗.(4.11) Moreover, 𝑓_𝑇1𝑝≈inf_𝑗𝜆_𝑗∶𝑓=_𝑗𝜆_𝑗𝑎_𝑗,(4.12) where the infimum is taken over all possible atomic decompositions of 𝑓∈𝑇¹_𝑝.

Note that the right hand side of (4.12) in fact defines a norm and then 𝑇¹_𝑝 becomes a Banach space.

Proof. Suppose that 𝑎 is a 𝑇¹_𝑝-atom on 𝕌^𝑛. Then there exists a ball 𝐵=𝐵(𝑧,𝑟)⊂ℍ^𝑛 such that supp 𝑎⊂𝑇(𝐵) and _𝑇(𝐵)𝑎𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤1|𝐵|^𝑝.(4.13) Fix 𝜀>0, and let 𝑤𝜉,𝜌=ℎ𝑟^{−(2𝑛+2)𝑝}𝑟min1,𝐷(𝜉,𝑧)^{(2𝑛+2)𝑝+𝜀},(4.14) where 𝐷(𝜉,𝑧) denotes the distance between (𝜉,𝜌) and (𝑧,0) in 𝕌^𝑛, 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∈ℍ^𝑛, the distance in 𝕌^𝑛 from the cone Γ(𝜉) to (𝑧,0) is 𝑐|𝑧⁻¹𝜉|=𝑐𝑑(𝑧,𝜉). Then 𝑁(𝑤)(𝑧)≤ℎ𝑟^{−(2𝑛+2)𝑝}𝑟min1,𝑐𝑑(𝜉,𝑧)^{(2𝑛+2)𝑝+𝜀}.(4.15) Therefore, ℎ⁻¹_ℍ𝑛𝑁(𝑤)𝑑Λ_𝑝^(∞)=^∞₀Λ_𝑝^(∞)𝜉∈ℍ^𝑛∶ℎ⁻¹𝑁(𝑤)≤>𝜆𝑑𝜆_{𝑟−(2𝑛+2)𝑝0}Λ_𝑝^(∞)𝐵1𝑧,𝑐𝑟^𝜀𝜆^{1/(2𝑛+2)𝑝+𝜀}𝑑𝜆≤𝐶^{𝑟−(2𝑛+2)𝑝}₀Λ_𝑝^(∞)𝐵1𝑧,𝑐𝑟^𝜀𝜆^{1/(2𝑛+2)𝑝+𝜀}𝑟𝑑𝜆≤𝐶^{𝜀/(2𝑛+2)𝑝+𝜀(2𝑛+2)𝑝𝑟−(2𝑛+2)𝑝}₀𝜆^{−(2𝑛+2)𝑝/((2𝑛+2)𝑝+𝜀)}𝑑𝜆=𝐶.(4.16) Thus, ∫_ℍ𝑛𝑁(𝑤)𝑑Λ_𝑝^(∞)≤1 by choosing ℎ=𝐶⁻¹. On the other hand, let 𝑤⁻¹=𝑟^{(2𝑛+2)𝑝} on 𝑇(𝐵). We have _𝑇(𝐵)𝑎𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}=𝑟^{(2𝑛+2)𝑝}_𝑇(𝐵)𝑎𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤𝑟^{(2𝑛+2)𝑝}|𝐵|^−𝑝=𝐶.(4.17) Therefore, 𝑎∈𝑇¹_𝑝 and ‖𝑎‖_𝑇¹_𝑝≤𝐶.
Now, taking a sum ∑_𝑗𝜆_𝑗𝑎_𝑗, where ∑_𝑗|𝜆_𝑗|<∞ and every 𝑎_𝑗 is 𝑇¹_𝑝-atom on 𝕌^𝑛, by Lemma 4.4, the sum converges in the quasinorm to 𝑓∈𝑇¹_𝑝 with 𝑓_𝑇1𝑝≤_𝑗𝜆_𝑗𝑎_{𝑗𝑇1𝑝}≤𝐶_𝑗𝜆_𝑗<∞.(4.18) That means 𝑓=∑_𝑗𝜆_𝑗𝑎_𝑗∈𝑇¹_𝑝.
Conversely, let 𝑓∈𝑇¹_𝑝. We choose a Borel measurable function 𝑤≥0 on 𝕌^𝑛 such that (4.3) holds and _𝕌𝑛𝑓𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤2𝑓²_𝑇1𝑝.(4.19)
For each 𝑘∈ℤ, let 𝐸_𝑘={𝑍∈ℍ^𝑛∶𝑁(𝑤)(𝑧)>2^𝑘}. By Lemma 2.4 and the definition of Λ_𝑝^(∞), it follows that there exists a disjoint dyadic cubes sequence {𝐼_𝑘,𝑗} such that _𝑗𝐼_𝑘,𝑗^𝑝≤2Λ_𝑝^(∞)𝐸_𝑘,𝑇𝐸_𝑘⊂_𝑗𝑆^∗𝐼_𝑘,𝑗,(4.20) where 𝑆^∗(𝐼_𝑘,𝑗)={(𝜉,𝜌)∈𝕌^𝑛∶𝜉∈𝐼_𝑘,𝑗,𝜌<2diam(𝐼_𝑘,𝑗)}. Define 𝑇_𝑘,𝑗=𝑆^∗𝐼_𝑘,𝑗⧵_{𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙.(4.21) Then 𝑇_𝑘,𝑗 is disjoint in 𝕌^𝑛 for different 𝑗,𝑘. Therefore, ^𝐾_{𝑘=−𝐾𝑗}𝑇_𝑘,𝑗=_𝑗𝑆^∗𝐼_−𝐾,𝑗⧵_{𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙⊃𝑇𝐸_−𝐾⧵_{𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙.(4.22) Note that ⋃_𝑘𝑇(𝐸_𝑘)={(𝜉,𝜌)∈𝕌^𝑛∶𝑤(𝜉,𝜌)>0}, (𝜉,𝜌)∉𝑇(𝐸_𝑘) implies 𝑤(𝜉,𝜌)≤2^𝑘. And each 𝑆^∗(𝐼_𝑚,𝑙) is contained in a cube of sidelength 4diam(𝐼_𝑚,𝑙) in 𝕌^𝑛. By using the subadditivity of the 𝑝-Hausdorff capacity and (4.3), there is Λ_𝑝^(∞)_{𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙≤𝐶_{𝑚>𝑘𝑙}𝐼_𝑚,𝑙^𝑝≤𝐶_𝑚>𝑘Λ_𝑝^(∞)𝐸_𝑚→0(4.23) as 𝑘→∞. Thus, by (4.21), _𝑘𝑗𝑇_𝑘,𝑗⊃_𝑘𝑇𝐸_𝑘⧵_{𝑘𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙=𝜉,𝜌∈𝕌^𝑛∶𝑤𝜉,𝜌>0⧵𝐸_∞,(4.24) where 𝐸_∞ is a set of zero 𝑝-Hausdorff capacity, that is, Λ_𝑝^(∞)𝐸_∞=Λ_𝑝^(∞)_{𝑘𝑚>𝑘𝑙}𝑆^∗𝐼_𝑚,𝑙=0.(4.25) Since 𝑤 is allowed to vanish only where 𝑓 vanishes, we have 𝑓=∑_𝑘,𝑗𝑓⋅𝜒_{𝑇𝑘,𝑗} a.e. on 𝕌^𝑛. Let 𝑎_𝑘,𝑗=𝐼^∗_𝑘,𝑗^𝑝_{𝑇𝑘,𝑗}𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)−1/2}𝑓⋅𝜒_{𝑇𝑘,𝑗},𝜆_𝑘,𝑗=𝐼^∗_𝑘,𝑗^𝑝_{𝑇𝑘,𝑗}𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2},(4.26) where the 𝐼^∗_𝑘,𝑗 is a cube as Lemma 2.4, that is, 𝐼^∗_𝑘,𝑗=5𝑐_𝑛𝐼_𝑘,𝑗.
Thus 𝑓=_𝑘,𝑗𝜆_𝑘,𝑗𝑎_𝑘,𝑗,a.e.on𝕌^𝑛.(4.27) Note that, by (4.21), 𝑇_𝑘,𝑗⊂𝑆^∗(𝐼_𝑘,𝑗)⊂𝑇(𝐵_𝑘,𝑗), where 𝐵_𝑘,𝑗 is the ball with the same center as 𝐼_𝑘,𝑗 and radius 𝑐𝑙(𝐼^∗_𝑘,𝑗) (𝑐>1 is a constant). Thus, 𝑎_𝑘,𝑗 is supported in 𝑇(𝐵_𝑘,𝑗), and _{𝑇(𝐵𝑘,𝑗)}𝑎_𝑘,𝑗𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}=𝐼^∗_𝑘,𝑗^𝑝_{𝑇𝑘,𝑗}𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)−1}_{𝑇(𝐵𝑘,𝑗)}𝑓⋅𝜒_{𝑇𝑘,𝑗}²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}=𝐼^∗_𝑘,𝑗^−𝑝=𝐶𝐵_𝑘,𝑗^−𝑝.(4.28) This means that every 𝑎_𝑘,𝑗 is a 𝑇¹_𝑝-atom. The remaining is to estimate ∑_𝑘,𝑗|𝜆_𝑘,𝑗|.
Notice the fact that 𝑤≤2^𝑘+1on𝑇_𝑘,𝑗⊂_𝑙𝑆^∗𝐼_𝑘+1,𝑙^𝑐⊂𝑇𝐸_𝑘+1^𝑐.(4.29) By the Cauchy-Schwarz inequality, (4.19) and (4.20), we have _𝑘,𝑗𝜆_𝑘,𝑗≤_𝑘,𝑗2^(𝑘+1)/2𝐼^∗_𝑘,𝑗^𝑝/2_{𝑇𝑘,𝑗}𝑓𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2}≤_𝑘,𝑗2^𝑘+1𝐼^∗_𝑘,𝑗^𝑝1/2_{𝑘,𝑗𝑇𝑘,𝑗}𝑓𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2}≤𝐶𝑓_{𝑇1𝑝𝑘}2^𝑘_𝑗𝐼_𝑘,𝑗^𝑝1/2≤𝐶𝑓_{𝑇1𝑝𝑘}2^𝑘Λ_𝑝^(∞)𝐸_𝑘^1/2=𝐶𝑓_𝑇1𝑝0^+∞Λ_𝑝^(∞)({𝜉∈ℍ^𝑛∶𝑁(𝑤)>𝜆})𝑑𝜆)^1/2≤𝐶𝑓_{𝑇1𝑝ℍ𝑛}𝑁(𝑤)𝑑Λ_𝑝^(∞)1/2≤𝐶𝑓_𝑇1𝑝.(4.30) If 𝑓=∑_𝑘𝜆_𝑘𝑎_𝑘 and every 𝑎_𝑘 is a 𝑇¹_𝑝-atom, then, by Lemma 4.4, we obtain 𝑓_𝑇1𝑝≤𝐶_𝑘𝜆_𝑘𝑎_{𝑘𝑇1𝑝}≤𝐶_𝑘𝜆_𝑘.(4.31) Thus 𝑓_𝑇1𝑝≈inf_𝑗𝜆_𝑗∶𝑓=_𝑗𝜆_𝑗𝑎_𝑗,(4.32) where the infimum is taken over all atomic decompositions of 𝑓∈𝑇¹_𝑝. The proof is complete.

The dual result is as follows.

Theorem 4.7. Let 𝑝∈(0,1]. Then the dual of 𝑇¹_𝑝 can be identified with 𝑇^∞_𝑝 under the pairing 𝑓,𝑔=_𝕌𝑛𝑓𝜉,𝜌𝑔𝜉,𝜌𝑑𝜉𝑑𝜌𝜌.(4.33)

Proof. We first show that _𝕌𝑛𝑓𝜉,𝜌𝑔𝜉,𝜌𝑑𝜉𝑑𝜌𝜌≤𝐶𝑓_𝑇1𝑝𝑔_𝑇∞𝑝(4.34) holds for all 𝑓∈𝑇¹_𝑝 and 𝑔∈𝑇^∞_𝑝.
In fact, assume that 𝑤 is a nonnegative Borel measurable function on 𝕌^𝑛 satisfying inequality (4.3) in Definition 4.2 and 𝑔∈𝑇^∞_𝑝. Then there exists a constant 𝐶 for any ball 𝐵⊂ℍ^𝑛 satisfying 1|𝐵|^𝑝_𝑇(𝐵)𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)}≤𝐶.(4.35) It means that |𝑔(𝜉,𝜌)|²𝜌^{−2(𝑛+1)(1−𝑝)−1}𝑑𝜉𝑑𝜌 is a 𝑝-Carleson measure, and ‖𝜇‖≈‖𝑔‖²_𝑇^∞_𝑝. Hence, by Theorem 3.1, we obtain _𝕌𝑛𝑤𝜉,𝜌𝑔𝜉,𝜌²𝜌^{−2(𝑛+1)(1−𝑝)−1}𝑑𝜉𝑑𝜌≤𝐶𝑔²_{𝑇∞𝑝ℍ𝑛}𝑁(𝑤)𝑑Λ_𝑝^(∞)≤𝐶𝑔²_𝑇∞𝑝.(4.36) Thus, for 𝑓∈𝑇¹_𝑝, by using the Cauchy-Schwarz inequality, there is _𝕌𝑛𝑓𝜉,𝜌𝑔𝜉,𝜌𝑑𝜉𝑑𝜌𝜌≤_𝕌𝑛𝑓²𝑤⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2}_𝕌^𝑛𝑔²𝑤𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)1/2}≤𝐶_𝕌𝑛𝑓𝜉,𝜌²𝑤𝜉,𝜌⁻¹𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)1/2}⋅𝑔_𝑇∞𝑝≤𝐶𝑓_𝑇1𝑝𝑔_𝑇∞𝑝.(4.37) This gives (4.34). Thus, every 𝑔∈𝑇^∞_𝑝 induces a bounded linear functional on 𝑇¹_𝑝 via the pairing (4.33). It suffices to prove the converse.
Let 𝐿 be a bounded linear functional on 𝑇¹_𝑝 and fix a ball 𝐵=𝐵(𝑧,𝑟)⊂ℍ^𝑛. If 𝑓 is supported in 𝑇(𝐵) with 𝑓∈𝐿²(𝑇(𝐵),𝜌⁻¹𝑑𝜉𝑑𝜌), then _𝑇(𝐵)𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}≤𝑟^{2(𝑛+1)(1−𝑝)}_𝑇(𝐵)𝑓𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌=𝐶𝑟^2(𝑛+1)|𝐵|^−𝑝𝑓²_{𝐿2(𝑇(𝐵),𝜌−1𝑑𝜉𝑑𝜌)}.(4.38) Therefore, 𝑓 is a multiple of a 𝑇¹_𝑝-atom with 𝑓²_𝑇1𝑝≤𝐶𝑟^2(𝑛+1)𝑓²_{𝐿2(𝑇(𝐵),𝜌−1𝑑𝜉𝑑𝜌)}.(4.39) Hence, 𝐿 induces a bounded linear functional on 𝐿²(𝑇(𝐵),𝜌⁻¹𝑑𝜉𝑑𝜌). Thus, there exists a function 𝑔 which is locally in 𝐿²(𝕌^𝑛,𝜌⁻¹𝑑𝜉𝑑𝜌) such that 𝐿𝑓=_𝕌𝑛𝑓𝜉,𝜌𝑔𝜉,𝜌𝑑𝜉𝑑𝜌𝜌,(4.40) whenever 𝑓∈𝑇¹_𝑝 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 𝑇¹_𝑝. Therefore, the rest of the proof is to show that 𝑔∈𝑇^∞_𝑝 and ‖𝑔‖_𝑇^∞_𝑝≤𝐶‖𝐿‖.
Now, again fix a ball 𝐵⊂ℍ^𝑛 and for every 𝜀>0, let 𝑓_𝜀𝜉,𝜌=𝜌^{−2(𝑛+1)(1−𝑝)}𝑔𝜉,𝜌𝜒_{𝑇𝜀(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)𝑓_𝜀𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1−2(𝑛+1)(1−𝑝)}=_{𝑇𝜀(𝐵)}𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)}<∞.(4.42) Hence, 𝑓_𝜀 is a multiple of a 𝑇¹_𝑝-atom with 𝑓_𝜀²_𝑇1𝑝≤𝐶𝑟^{2(𝑛+1)𝑝}_{𝑇𝜀(𝐵)}𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)},(4.43) where 𝐶 is independent of 𝜀. By (4.40), we obtain _{𝑇𝜀(𝐵)}𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)}=𝐿𝑓_𝜀≤‖𝐿‖⋅𝑓_{𝜀𝑇1𝑝}≤𝐶‖𝐿‖|𝐵|^𝑝_{𝑇𝜀(𝐵)}𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)1/2}.(4.44) Thus |𝐵|^−𝑝_{𝑇𝜀(𝐵)}𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)1/2}≤𝐶‖𝐿‖.(4.45) It follows that |𝐵|^−𝑝_𝑇(𝐵)𝑔𝜉,𝜌²𝑑𝜉𝑑𝜌𝜌^{1+2(𝑛+1)(1−𝑝)1/2}≤𝐶‖𝐿‖,(4.46) since (4.40) is true for all 𝜀>0. Therefore, 𝑔∈𝑇^∞_𝑝 and ‖𝑔‖_𝑇^∞_𝑝≤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∈𝑇¹_𝑝. 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).