# The Calderón Reproducing Formula Associated with the Heisenberg GroupHd

Authors: Jinsen Xiao; Jianxun He
Journal: Mathematical Problems in Engineering
 (2010)

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## Abstract

We establish the Calderón reproducing formula for functions inL2 on the Heisenberg group Hd. Also, we develop this formula in Lp(Hd) with 1<p<∞.


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## Body

## 1. Introduction

The classical Calderón reproducing formula reads(1.1)f=∫0+∞f*ϕt*ψtdtt,
where ϕt(x)=t-1ϕ(x/t), ψt(x)=t-1ψ(x/t), and * denotes the convolution on R. The Calderón reproducing formula is a useful tool in pure and applied mathematics (see [1–4]), particularly in wavelet theory (see [5, 6]). We always call (1.1) an inverse formula of wavelet transform. In [7], the authors generalized (1.1) to Rn when ϕ and ψ are sufficiently nice normalized radial wavelet functions. The generalization of (1.1) involving nonradial wavelets ϕ and ψ can be written in the following form:(1.2)f=∫SO(n)dγ∫0+∞f*ϕγ,t*ψγ,tdtt,
where ϕγ,t and ψγ,t are rotated versions of ϕ and ψ on Rn. The authors in [8, 9] established (1.1) for f∈Lp(R). Holschneider [10] studied the formula (1.2) in case f∈Lp(R2) and gave an inversion formula of the Radon transform in Lp-space by using wavelets. Furthermore, Rubin [4] developed the Calderón reproducing formula, windowed X-ray transforms, the Radon transforms, and k-plane transforms in Lp-spaces on Rn.It is a remarkable fact that the Heisenberg group, denoted byHd, arises in two fundamental but different setting in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental nations of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in connection with the Fourier transform, pseudodifferential operators, and related matters (see [11]). Due to this reason, many interesting works were devoted to the theory of harmonic analysis on Hd in [11–13] and the references therein. Also, the researches of wavelet analysis on Hd are concerned increasingly; for this we refer readers to [14–16]. And the inversion formula of the Radon transform by using inverse wavelet transform on Hd was established in [17]. Our goal of the present article is to study the Calderón reproducing formula on the Heisenberg group in Lp-space with 1<p<∞. In the sequel we will develop the theory of inverse Radon transform on Hd.The Heisenberg groupHd is a Lie group with the underlying manifold Cd×R, and the multiplication law is given by(1.3)(z,t)(z′,t′)=(z+z′,t+t′+2Imzz′¯),
where zz′¯=∑j=1dzjzj′¯. The dilation of Hd is defined by ρ(z,t)=(ρz,ρ2t) with ρ>0. For (z,t)∈Hd, the homogeneous norm of (z,t) is given by (see [11]) (1.4)|(z,t)|=|(z,t)-1|=max{|z|,|t|1/2}.
Notice that |ρ(z,t)|=max{|ρz|,|ρ2t|1/2}=ρ|(z,t)|. In addition, |·| satisfies the quasitriangle inequality:(1.5)|(z,t)(z′,t′)|≤|(z,t)|+|(z′,t′)|.
The homogeneous dimension of Hd is 2d+2, and the volume of a ball ℬ((z,t),r)={(z′,t′)∈Hd:|(z,t)-1(z′,t′)|≤r} is c′r2d+2, where c′ is a constant.LetP={(z,t,ρ):(z,t)∈Hd,ρ>0}; then P is a locally compact nonunimodular group with the group law (1.6)(z,t,ρ)(z′,t′,ρ′)=(z+ρz′,t+ρ2t′+2ρImzz′¯,ρρ′).
The left and right Haar measures on P are given by (1.7)dμl(z,t,ρ)=dzdtdρρ2d+3,dμr(z,t,ρ)=dzdtdρρ,
where dz denotes the Lebesgue measure on Cd.LetLp(Hd) be the space of measurable functions f on Hd, such that (1.8)∥f∥Lp(Hd)=(∫Hd|f(z,t)|pdzdt)1/p<+∞,ifp∈[1,+∞),∥f∥Lp(Hd)=esssup(z,t)∈Hd|f(z,t)|<+∞,ifp=+∞.LetZ+={0,1,2,…} and α=(α1,α2,…,αd)∈(Z+)d. The Fock space ℋ is the space of holomorphic functions F on Cd such that (1.9)∥F∥ℋ2=(2π)d∫Cd|F(ζ)|2e-2|ζ|2dζ<+∞.
From [18] we know that {Eα(ζ)=(2ζ)α/α!:α∈(Z+)d} is an orthonormal basis of the Hilbert space ℋ. For λ∈R∖{0}, let πλ be the Bargmann-Fock representation of Hd which acts on ℋ by (1.10)πλ(z,t)F(ζ)={e-iλt-λ|z|2+2λζz¯F(ζ-λz),ifλ>0,e-iλt+λ|z|2-2|λ|ζzF(ζ+λz¯),ifλ<0.
The group Fourier transform of a function f∈L1(Hd) is defined by(1.11)f̂(λ)=∫Hdf(z,t)πλ(z,t)dzdt.LetSp(ℋ)(1≤p<+∞) be the classes of Schatten-von Neumann operators on Hilbert space ℋ, and let S∞(ℋ) denote the algebra of all bounded operators, that is, S∞(ℋ)=B(ℋ). For T∈Sp(ℋ), let ∥T∥p=(tr(T*T)1/p)p/2 denote the Sp-norm of T. If p=2,∥T∥2 is just the Hilbert-Schmit norm of T, that denotes ∥T∥HS. Let ∥T∥∞ denote the usual operator norm of T in S∞(ℋ). For 1≤p≤+∞, let ℒp be the Banach space consisting of all weak measurable operator value functions F, which also satisfy F(λ)∈Sp(ℋ|λ|), a.e. λ∈R∖{0}, and (1.12)∥F∥ℒp=(1πd+1∫R∖{0}∥F(λ)∥pp|λ|ddλ)1/p<+∞,if1≤p<+∞,∥F∥ℒp=esssupλ∈R∖{0}∥F(λ)∥∞<+∞,ifp=∞.
For f,g∈L2(Hd), the Parseval formula is(1.13)〈f,g〉L2(Hd)=2d-1πd+1∫-∞+∞tr(ĝ(λ)*f̂(λ))|λ|ddλ,
where ĝ(λ)* denotes the adjoint of ĝ(λ). The Plancherel formula is(1.14)∥f∥L2(Hd)2=2d-1πd+1∫-∞+∞∥f̂(λ)∥HS2|λ|ddλ.
As a consequence of (1.13), one has the inversion of the Fourier transform:(1.15)f(z,t)=2d-1πd+1∫-∞+∞tr(πλ(z,t)*f̂(λ))|λ|ddλ.
Suppose ρ>0, and let(1.16)fρ(z,t)=ρ-(2d+2)f(ρ-1(z,t))=ρ-(2d+2)f(zρ,tρ2).
By a direct computation, we have(1.17)f̂ρ(λ)=f̂(ρλ).
Let f*g be the convolution of f and g, that is,(1.18)f*g(z,t)=∫Hdf(z′,t′)g((z′,t′)-1(z,t))dz′dt′.
Then(1.19)f*ĝ(λ)=f̂(λ)ĝ(λ).
We should notice the following facts: if f̃(z,t)=f((z,t)-1)¯=f(-z,-t)¯, then(1.20)f̃̂(λ)=f̂(λ)*.
And if g(z,t)=f(-z¯,-t), then(1.21)ĝ(λ)=f̂(-λ).
The further detail of harmonic analysis on Hd can be found in [11, 12].


## 2. Calderón Reproducing Formula

The authors in [14, 15, 18] studied the theory of continuous wavelet associated with the concept of square integrable group representations. The unitary representation of P on L2(Hd) is defined by (2.1)U(z,t,ρ)f(z′,t′)=ρ-(d+1)f(z′-zρ,t′-t-2Imzz′¯ρ2).
Let R+ denote the set of all positive real numbers, R-=-R+. Let Pα(α∈(Z+)d) be the projection from L2(Rd) to 1-dimensional subspace spanned by Eα, and let σ=+or-,(2.2)Hασ={f∈L2(Hd):f̂(λ)=f̂(λ)Pα,andf̂(λ)=0ifλ∉Rσ}.
From [15, Theorem 1], we have(2.3)L2(Hd)=⨁α∈(Z+)d(Hα+⊕Hα-).Letα∈(Z+)d and σ=+or-, ϕ∈Hασ; if ϕ≠0 and satisfies(2.4)Cϕ=〈∫Rσϕ̂(λ)*ϕ̂(λ)dλ|λ|Eα,Eα〉ℋ<+∞,
then we call ϕ an admissible wavelet and write ϕ∈AWασ. Let ϕ∈AWασ,f∈Hασ; the continuous wavelet transform of f with respect to ϕ is defined by(2.5)Wϕf(z,t,ρ)=〈f,U(z,t,ρ)ϕ〉L2(Hd).
And the following Calderón reproducing formula holds in the weak sense:(2.6)f(z′,t′)=Cϕ-1∫0+∞∫HdWϕf(z,t,ρ)U(z,t,ρ)ϕ(z′,t′)dzdtdρρ2d+3.


### 2.1. Calderón Reproducing Formula inL2(Hd)

By (1.16) and (1.18), we can rewrite (2.5) and (2.6) as follows:(2.7)Wϕf(z,t,ρ)=ρd+1f*ϕρ̃(z,t),(2.8)f(z,t)=Cϕ-1∫0+∞f*ϕρ*ϕρ̃(z,t)dρρ.For0<ɛ<η<+∞, let(2.9)fɛ,η(z,t)=Cϕ-1∫ɛηf*ϕρ*ϕρ̃(z,t)dρρ,(2.10)Φɛ,η(z,t)=Cϕ-1∫ɛηϕρ*ϕρ̃(z,t)dρρ,
then fɛ,η(z,t)=f*Φɛ,η(z,t). We are now in a position to show that fɛ,η converges to f in L2-space when ɛ→0 and η→∞. The result in this paper is an extension of that of Mourou and Trimèche [19].Lemma 2.1.
Suppose thatϕ∈AWασ and ϕ∈Hασ satisfies ϕ̂(λ)∈S∞(ℋ). Let Φɛ,η be defined by (2.10). Then one has Φɛ,η∈L2(Hd).Proof.
By Hölder's inequality, we have(2.11)|Φɛ,η|2≤Cϕ-2∫ɛη|ϕρ*ϕρ̃(z,t)|2dρρ∫ɛηdρρ.
Thus
(2.12)∫Hd|Φɛ,η|2dzdt≤Cϕ-2∫ɛη∫Hd|ϕρ*ϕρ̃(z,t)|2dzdtdρρ∫ɛηdρρ.
By (1.14), (1.19), and (1.20), we have
(2.13)∫Hd|ϕρ*ϕρ̃(z,t)|2dzdt=2d-1πd+1∫-∞+∞∥ϕ̂(ρλ)ϕ̂(ρλ)*∥HS2|λ|ddλ=2d-1πd+1∫-∞+∞tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ.
Noticing that
(2.14)tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))=∑α〈ϕ̂(ρλ)*ϕ̂(ρλ)Eα,ϕ̂(ρλ)*ϕ̂(ρλ)Eα〉ℋ,
we obtain
(2.15)tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))≤∥ϕ̂(ρλ)∥∞2tr(ϕ̂(ρλ)*ϕ̂(ρλ)).
Thus we get
(2.16)∫Hd|ϕρ*ϕρ̃(z,t)|2dzdt≤2d-1πd+1∫-∞+∞∥ϕ̂∥ℒ∞2tr(ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ=2d-1πd+1∥ϕ̂∥ℒ∞2∫-∞+∞tr(ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ=∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2.
Therefore,
(2.17)∫Hd|Φɛ,η|2dzdt≤Cϕ-2∫ɛη∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2dρρ∫ɛηdρρ=Cϕ-2∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2∫ɛηρ-2(d+1)dρρ∫ɛηdρρ<+∞.
Then we complete the proof of this lemma.Theorem 2.2.
Letϕ∈AWασ and ϕ∈Hασ satisfy ϕ̂(λ)∈S∞(ℋ). Then for f∈Hασ, one has fɛ,η∈L2(Hd) and limɛ→0,η→∞fɛ,η=f.Proof.
Notice that(2.18)fɛ,η(z,t)=f*Φɛ,η(z,t),
and by (1.19) and Lemma 2.1 we deduce fɛ,η∈L2(Hd). Then by (1.17) and (1.19), we have
(2.19)limɛ→0η→∞〈∫Hdfɛ,η(z,t)πλ(z,t)dzdtEα,Eα〉ℋ=limɛ→0η→∞〈∫Hdf*Φɛ,η(z,t)πλ(z,t)dzdtEα,Eα〉ℋ=limɛ→0η→∞〈Cϕ-1∫ɛηf̂(λ)ϕ̂(ρλ)*ϕ̂(ρλ)dρρEα,Eα〉ℋ=limɛ→0η→∞Cϕ-1〈∫ɛηϕ̂ασ(λ′)*ϕ̂ασ(λ′)dλ′|λ′|Eα,Eα〉ℋ〈f̂ασ(λ)Eα,Eα〉ℋ=〈f̂(λ)Eα,Eα〉ℋ,
where σ=+ if λ>0, otherwise σ=-. By (1.11) we get the desired result.


### 2.2. Calderón Reproducing Formula inLp(Hd) with 1<p<∞

Forf∈Lp(Hd) with 1<p<∞, the continuous wavelet transform of f with respect to a wavelet ϕ can be defined by formula (2.7) under certain conditions on ϕ. In this part we will show that fɛ,η converges to f.Letf be a measurable function on Hd; for (z,t)∈Hd, define(2.20)f*(z,t)=esssup{|f(z′,t′)|:(z′,t′)∈Hd,|(z′,t′)|>|(z,t)|}.
It is easy to see that f* is nonnegative and radially decreasing, that is, (2.21)0≤f*(z1,t1)≤f*(z2,t2)onlyif|(z1,t1)|>|(z2,t2)|,
and f*≥|f| a.e. on Hd.Letg∈L1(R+). Then for any (z,t)∈Hd, we define(2.22)G*(z,t)=sup{ρ-(2d+2)|∫ρ+∞g(s)ds|:ρ>|(z,t)|},
and thus we have the following lemma.Lemma 2.3.
Letg∈L1(R+) and G*(z,t) be defined by (2.22). Then one has
(2.23)∫Hd∖ℬG*(z,t)dzdt≤c∫1+∞|g(s)|lnsds,
where c is a positive constant, ℬ=ℬ((0,0),1).Proof.
First we let(2.24)G(z,t)=|(z,t)|-(2d+2)∫|(z,t)|+∞|g(s)|ds.
It is obvious that G*(z,t)≤G(z,t) for any (z,t)∈Hd∖{(0,0)}. Since G(z,t) is nonnegative and radially decreasing, from [11, page 542], we know that there exists a positive constant c such that
(2.25)∫HdG(z,t)dzdt=c∫0+∞G(r)r2d+1dr.
Thus,
(2.26)∫Hd∖ℬG*(z,t)dzdt≤∫Hd∖ℬG(z,t)dzdt=c∫1+∞G(r)r2d+1dr=c∫1+∞r-(2d+2)∫r+∞|g(s)|dsr2d+1dr=c∫1+∞∫1s|g(s)|dsdrr=c∫1+∞|g(s)|lnsds.
This completes the proof.Letk be a radial function in Lp(Hd) (1<p<∞) and let(2.27)K(z,t)=|(z,t)|-(2d+2)∫0|(z,t)|k(sω)s2d+1ds
for any (z,t)∈Hd∖{(0,0)}, where ω is a unit vector of Hd. Then we have the following lemma.Lemma 2.4.
Letk be a radial function in Lp(Hd)(1<p<∞) and let K(z,t) be defined by k in (2.27). If k satisfies ∫Hdk(z,t)dzdt=0, and k(z,t)ln|(z,t)|∈L1(Hd), then K∈L1(Hd).Proof.
Analogous to (2.20) we define
(2.28)K*(z,t)=esssup{|K(z′,t′)|:(z′,t′)∈Hd,|(z′,t′)|>|(z,t)|},
and then K*≥|K| a.e. on Hd.
Let(z,t),(z′,t′)∈Hd and 0<|(z,t)|<|(z′,t′)|; by (2.25) we have
(2.29)|K(z',t')|=1|(z′,t′)|(2d+2)|∫0|(z′,t′)|k(sω)s2d+1ds|≤1|(z′,t′)|(2d+2)∫0|(z',t')||k(sω)|s2d+1ds=1c|(z′,t′)|(2d+2)∫ℬ((0,0),|(z',t')|)|k(z′′,t′′)|'dz′′dt′′≤22d+2c(|(z,t)|+|(z′,t′)|)(2d+2)∫ℬ((z,t),|(z,t)|+|(z',t')|)|k(z′′,t′′)|dz′′dt′′≤22d+2(Mk)(z,t),
where Mk is the Hardy-Littlewood maximal function of k. From the definition of K*, we have K*≤22d+2(Mk). Because k∈Lp(Hd), we get K*∈Lp(Hd).
By the hypothesisk is radial and ∫Hdk(z,t)dzdt=0; together with the definition of K, we have
(2.30)|K(z,t)|=1|(z,t)|(2d+2)|∫|(z,t)|+∞k(sω)s2d+1ds|.
Since k(z,t)(ln|(z,t)|)∈L1(Hd), it follows from Lemma 2.3 that ∫Hd∖ℬK*(z,t)dzdt<+∞, that is, K*∈Lloc1(Hd). On the other hand, K*∈Lp(Hd)⊂Lloc1(Hd), and thus K*∈L1(Hd), which implies that K∈L1(Hd).Without loss of generality, we assume thatCϕ=1; then (2.9) and (2.10) can be written as(2.31)Φɛ,η(z,t)=∫ɛηϕρ*ϕρ̃(z,t)dρρ,(2.32)fɛ,η(z,t)=∫ɛηf*ϕρ*ϕρ̃(z,t)dρρ.In fact,Φɛ,η is always stated under conditions on k:=ϕ*ϕ̃ rather than under conditions on ϕ for convenience (see [4, 10]). By Lemma 2.4 we have the following theorem.Theorem 2.5.
Letk be in the conditions of Lemma 2.4 and let K be defined by (2.27). Suppose ϕ*ϕ̃=k and ∫HdK(z,t)dzdt=1. Then for f∈Lp(Hd), one has limɛ→0,η→∞fɛ,η=f.Proof.
From (2.31) we have
(2.33)Φɛ,η(z,t)=Φɛ,∞(z,t)-Φη,∞(z,t).
Since k(z,t)=ϕ*ϕ̃(z,t), and ϕ̃(z,t)=ϕ(-z,-t)¯, we deduce
(2.34)kρ(z,t)=ρ-(2d+2)∫Hdϕ(z′,t′)ϕ¯((ρ-1z,ρ-2t)-1(z′,t′))dz′dt′=ρ-(4d+4)∫Hdϕ(z′,t′)ϕ¯(ρ-1((z,t)-1(ρz′,ρ2t′)))d(ρz′)d(ρ2t')=ρ-(4d+4)∫Hdϕ(ρ-1(z′,t′))ϕ¯(ρ-1((z,t)-1(z′,t′)))dz′dt′=ϕρ*ϕρ̃(z,t).
Then we have
(2.35)Φɛ,∞(z,t)=∫ɛ+∞kρ(z,t)dρρ=∫ɛ+∞k((z,t)ρ)dρρ2d+3=1|(z,t)|2d+2∫ɛ/|z,t|+∞k((z,t)ρ|(z,t)|)dρρ2d+3=1|(z,t)|2d+2∫0|z,t|/ɛk(ρ(z,t)|(z,t)|)ρ2d+1dρ=ɛ-2(d+1)|(z,t)/ɛ|2d+2∫0|(z,t)|/ɛk(ρω)ρ2d+1dρ=Kɛ(z,t).
By Lemma 2.4 together with the approximation of the identity, we have
(2.36)limɛ→0η→∞fɛ,η(z,t)=limɛ→0η→∞Φɛ,η*f(z,t)=limɛ→0Kɛ*f(z,t)-limη→∞Kη*f(z,t)=f(z,t).
Then we complete the proof of this theorem.


## 2.1. Calderón Reproducing Formula inL2(Hd)

By (1.16) and (1.18), we can rewrite (2.5) and (2.6) as follows:(2.7)Wϕf(z,t,ρ)=ρd+1f*ϕρ̃(z,t),(2.8)f(z,t)=Cϕ-1∫0+∞f*ϕρ*ϕρ̃(z,t)dρρ.For0<ɛ<η<+∞, let(2.9)fɛ,η(z,t)=Cϕ-1∫ɛηf*ϕρ*ϕρ̃(z,t)dρρ,(2.10)Φɛ,η(z,t)=Cϕ-1∫ɛηϕρ*ϕρ̃(z,t)dρρ,
then fɛ,η(z,t)=f*Φɛ,η(z,t). We are now in a position to show that fɛ,η converges to f in L2-space when ɛ→0 and η→∞. The result in this paper is an extension of that of Mourou and Trimèche [19].Lemma 2.1.
Suppose thatϕ∈AWασ and ϕ∈Hασ satisfies ϕ̂(λ)∈S∞(ℋ). Let Φɛ,η be defined by (2.10). Then one has Φɛ,η∈L2(Hd).Proof.
By Hölder's inequality, we have(2.11)|Φɛ,η|2≤Cϕ-2∫ɛη|ϕρ*ϕρ̃(z,t)|2dρρ∫ɛηdρρ.
Thus
(2.12)∫Hd|Φɛ,η|2dzdt≤Cϕ-2∫ɛη∫Hd|ϕρ*ϕρ̃(z,t)|2dzdtdρρ∫ɛηdρρ.
By (1.14), (1.19), and (1.20), we have
(2.13)∫Hd|ϕρ*ϕρ̃(z,t)|2dzdt=2d-1πd+1∫-∞+∞∥ϕ̂(ρλ)ϕ̂(ρλ)*∥HS2|λ|ddλ=2d-1πd+1∫-∞+∞tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ.
Noticing that
(2.14)tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))=∑α〈ϕ̂(ρλ)*ϕ̂(ρλ)Eα,ϕ̂(ρλ)*ϕ̂(ρλ)Eα〉ℋ,
we obtain
(2.15)tr(ϕ̂(ρλ)*ϕ̂(ρλ)ϕ̂(ρλ)*ϕ̂(ρλ))≤∥ϕ̂(ρλ)∥∞2tr(ϕ̂(ρλ)*ϕ̂(ρλ)).
Thus we get
(2.16)∫Hd|ϕρ*ϕρ̃(z,t)|2dzdt≤2d-1πd+1∫-∞+∞∥ϕ̂∥ℒ∞2tr(ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ=2d-1πd+1∥ϕ̂∥ℒ∞2∫-∞+∞tr(ϕ̂(ρλ)*ϕ̂(ρλ))|λ|ddλ=∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2.
Therefore,
(2.17)∫Hd|Φɛ,η|2dzdt≤Cϕ-2∫ɛη∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2dρρ∫ɛηdρρ=Cϕ-2∥ϕ̂∥ℒ∞2∥ϕρ∥L2(Hd)2∫ɛηρ-2(d+1)dρρ∫ɛηdρρ<+∞.
Then we complete the proof of this lemma.Theorem 2.2.
Letϕ∈AWασ and ϕ∈Hασ satisfy ϕ̂(λ)∈S∞(ℋ). Then for f∈Hασ, one has fɛ,η∈L2(Hd) and limɛ→0,η→∞fɛ,η=f.Proof.
Notice that(2.18)fɛ,η(z,t)=f*Φɛ,η(z,t),
and by (1.19) and Lemma 2.1 we deduce fɛ,η∈L2(Hd). Then by (1.17) and (1.19), we have
(2.19)limɛ→0η→∞〈∫Hdfɛ,η(z,t)πλ(z,t)dzdtEα,Eα〉ℋ=limɛ→0η→∞〈∫Hdf*Φɛ,η(z,t)πλ(z,t)dzdtEα,Eα〉ℋ=limɛ→0η→∞〈Cϕ-1∫ɛηf̂(λ)ϕ̂(ρλ)*ϕ̂(ρλ)dρρEα,Eα〉ℋ=limɛ→0η→∞Cϕ-1〈∫ɛηϕ̂ασ(λ′)*ϕ̂ασ(λ′)dλ′|λ′|Eα,Eα〉ℋ〈f̂ασ(λ)Eα,Eα〉ℋ=〈f̂(λ)Eα,Eα〉ℋ,
where σ=+ if λ>0, otherwise σ=-. By (1.11) we get the desired result.


## 2.2. Calderón Reproducing Formula inLp(Hd) with 1<p<∞

Forf∈Lp(Hd) with 1<p<∞, the continuous wavelet transform of f with respect to a wavelet ϕ can be defined by formula (2.7) under certain conditions on ϕ. In this part we will show that fɛ,η converges to f.Letf be a measurable function on Hd; for (z,t)∈Hd, define(2.20)f*(z,t)=esssup{|f(z′,t′)|:(z′,t′)∈Hd,|(z′,t′)|>|(z,t)|}.
It is easy to see that f* is nonnegative and radially decreasing, that is, (2.21)0≤f*(z1,t1)≤f*(z2,t2)onlyif|(z1,t1)|>|(z2,t2)|,
and f*≥|f| a.e. on Hd.Letg∈L1(R+). Then for any (z,t)∈Hd, we define(2.22)G*(z,t)=sup{ρ-(2d+2)|∫ρ+∞g(s)ds|:ρ>|(z,t)|},
and thus we have the following lemma.Lemma 2.3.
Letg∈L1(R+) and G*(z,t) be defined by (2.22). Then one has
(2.23)∫Hd∖ℬG*(z,t)dzdt≤c∫1+∞|g(s)|lnsds,
where c is a positive constant, ℬ=ℬ((0,0),1).Proof.
First we let(2.24)G(z,t)=|(z,t)|-(2d+2)∫|(z,t)|+∞|g(s)|ds.
It is obvious that G*(z,t)≤G(z,t) for any (z,t)∈Hd∖{(0,0)}. Since G(z,t) is nonnegative and radially decreasing, from [11, page 542], we know that there exists a positive constant c such that
(2.25)∫HdG(z,t)dzdt=c∫0+∞G(r)r2d+1dr.
Thus,
(2.26)∫Hd∖ℬG*(z,t)dzdt≤∫Hd∖ℬG(z,t)dzdt=c∫1+∞G(r)r2d+1dr=c∫1+∞r-(2d+2)∫r+∞|g(s)|dsr2d+1dr=c∫1+∞∫1s|g(s)|dsdrr=c∫1+∞|g(s)|lnsds.
This completes the proof.Letk be a radial function in Lp(Hd) (1<p<∞) and let(2.27)K(z,t)=|(z,t)|-(2d+2)∫0|(z,t)|k(sω)s2d+1ds
for any (z,t)∈Hd∖{(0,0)}, where ω is a unit vector of Hd. Then we have the following lemma.Lemma 2.4.
Letk be a radial function in Lp(Hd)(1<p<∞) and let K(z,t) be defined by k in (2.27). If k satisfies ∫Hdk(z,t)dzdt=0, and k(z,t)ln|(z,t)|∈L1(Hd), then K∈L1(Hd).Proof.
Analogous to (2.20) we define
(2.28)K*(z,t)=esssup{|K(z′,t′)|:(z′,t′)∈Hd,|(z′,t′)|>|(z,t)|},
and then K*≥|K| a.e. on Hd.
Let(z,t),(z′,t′)∈Hd and 0<|(z,t)|<|(z′,t′)|; by (2.25) we have
(2.29)|K(z',t')|=1|(z′,t′)|(2d+2)|∫0|(z′,t′)|k(sω)s2d+1ds|≤1|(z′,t′)|(2d+2)∫0|(z',t')||k(sω)|s2d+1ds=1c|(z′,t′)|(2d+2)∫ℬ((0,0),|(z',t')|)|k(z′′,t′′)|'dz′′dt′′≤22d+2c(|(z,t)|+|(z′,t′)|)(2d+2)∫ℬ((z,t),|(z,t)|+|(z',t')|)|k(z′′,t′′)|dz′′dt′′≤22d+2(Mk)(z,t),
where Mk is the Hardy-Littlewood maximal function of k. From the definition of K*, we have K*≤22d+2(Mk). Because k∈Lp(Hd), we get K*∈Lp(Hd).
By the hypothesisk is radial and ∫Hdk(z,t)dzdt=0; together with the definition of K, we have
(2.30)|K(z,t)|=1|(z,t)|(2d+2)|∫|(z,t)|+∞k(sω)s2d+1ds|.
Since k(z,t)(ln|(z,t)|)∈L1(Hd), it follows from Lemma 2.3 that ∫Hd∖ℬK*(z,t)dzdt<+∞, that is, K*∈Lloc1(Hd). On the other hand, K*∈Lp(Hd)⊂Lloc1(Hd), and thus K*∈L1(Hd), which implies that K∈L1(Hd).Without loss of generality, we assume thatCϕ=1; then (2.9) and (2.10) can be written as(2.31)Φɛ,η(z,t)=∫ɛηϕρ*ϕρ̃(z,t)dρρ,(2.32)fɛ,η(z,t)=∫ɛηf*ϕρ*ϕρ̃(z,t)dρρ.In fact,Φɛ,η is always stated under conditions on k:=ϕ*ϕ̃ rather than under conditions on ϕ for convenience (see [4, 10]). By Lemma 2.4 we have the following theorem.Theorem 2.5.
Letk be in the conditions of Lemma 2.4 and let K be defined by (2.27). Suppose ϕ*ϕ̃=k and ∫HdK(z,t)dzdt=1. Then for f∈Lp(Hd), one has limɛ→0,η→∞fɛ,η=f.Proof.
From (2.31) we have
(2.33)Φɛ,η(z,t)=Φɛ,∞(z,t)-Φη,∞(z,t).
Since k(z,t)=ϕ*ϕ̃(z,t), and ϕ̃(z,t)=ϕ(-z,-t)¯, we deduce
(2.34)kρ(z,t)=ρ-(2d+2)∫Hdϕ(z′,t′)ϕ¯((ρ-1z,ρ-2t)-1(z′,t′))dz′dt′=ρ-(4d+4)∫Hdϕ(z′,t′)ϕ¯(ρ-1((z,t)-1(ρz′,ρ2t′)))d(ρz′)d(ρ2t')=ρ-(4d+4)∫Hdϕ(ρ-1(z′,t′))ϕ¯(ρ-1((z,t)-1(z′,t′)))dz′dt′=ϕρ*ϕρ̃(z,t).
Then we have
(2.35)Φɛ,∞(z,t)=∫ɛ+∞kρ(z,t)dρρ=∫ɛ+∞k((z,t)ρ)dρρ2d+3=1|(z,t)|2d+2∫ɛ/|z,t|+∞k((z,t)ρ|(z,t)|)dρρ2d+3=1|(z,t)|2d+2∫0|z,t|/ɛk(ρ(z,t)|(z,t)|)ρ2d+1dρ=ɛ-2(d+1)|(z,t)/ɛ|2d+2∫0|(z,t)|/ɛk(ρω)ρ2d+1dρ=Kɛ(z,t).
By Lemma 2.4 together with the approximation of the identity, we have
(2.36)limɛ→0η→∞fɛ,η(z,t)=limɛ→0η→∞Φɛ,η*f(z,t)=limɛ→0Kɛ*f(z,t)-limη→∞Kη*f(z,t)=f(z,t).
Then we complete the proof of this theorem.



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*Source: 132186-2010-07-04.xml*