The Calderón Reproducing Formula Associated with the Heisenberg Group

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 R 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:


Introduction
The classical Calder ón reproducing formula reads where φ t x t −1 φ x/t , ψ t x t −1 ψ x/t , and * denotes the convolution on R. The Calder ón 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 R n 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: where φ γ,t and ψ γ,t are rotated versions of φ and ψ on R n .The authors in 8, 9 established 1.1 for f ∈ L p R .Holschneider 10 studied the formula 1.2 in case f ∈ L p R 2 and gave an inversion formula of the Radon transform in L p -space by using wavelets.Furthermore, Mathematical Problems in Engineering Rubin 4 developed the Calder ón reproducing formula, windowed X-ray transforms, the Radon transforms, and k-plane transforms in L p -spaces on R n .It is a remarkable fact that the Heisenberg group, denoted by H d , 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 H d in 11-13 and the references therein.Also, the researches of wavelet analysis on H d are concerned increasingly; for this we refer readers to 14 1.9 From 18 we know that The group Fourier transform of a function f ∈ L 1 H d is defined by Let S p 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 ∈ S p H , let T p tr T * T 1/p p/2 denote the S p -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 L p be the Banach space consisting of all weak measurable operator value functions F, which also satisfy F λ ∈ S p H |λ| , a.e.λ ∈ R \ {0}, and where g λ * denotes the adjoint of g λ .The Plancherel formula is As a consequence of 1.13 , one has the inversion of the Fourier transform: Suppose ρ > 0, and let By a direct computation, we have Let f * g be the convolution of f and g, that is, We should notice the following facts The further detail of harmonic analysis on H d can be found in 11, 12 .

Calder ón 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 L 2 H d is defined by Let R denote the set of all positive real numbers, R − −R .Let P α α ∈ Z d be the projection from L 2 R d to 1-dimensional subspace spanned by E α , and let σ or −, From 15, Theorem 1 , we have then we call φ an admissible wavelet and write φ ∈ AW σ α .Let φ ∈ AW σ α , f ∈ H σ α ; the continuous wavelet transform of f with respect to φ is defined by And the following Calder ón reproducing formula holds in the weak sense: tr φ ρλ * φ ρλ φ ρλ * φ ρλ |λ| d dλ.

2.17
Then we complete the proof of this lemma.

Calder ón Reproducing Formula in L p H d with 1 < p < ∞
For f ∈ L p H d 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 f ε,η converges to f.Let f be a measurable function on It is easy to see that f * is nonnegative and radially decreasing, that is, and thus we have the following lemma.
Lemma 2.3.Let g ∈ L 1 R and G * z, t be defined by 2.22 .Then one has where c is a positive constant, B B 0, 0 , 1 .
Proof.First we let

2.24
It is obvious that G * z, t ≤ G z, t for any z, t ∈ H d \ { 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 Thus, Without loss of generality, we assume that C φ 1; then 2.9 and 2.10 can be written as

2.32
In fact, Φ ε,η is always stated under conditions on k : φ * φ rather than under conditions on φ for convenience see 4, 10 .By Lemma 2.4 we have the following theorem.Theorem 2.5.Let k be in the conditions of Lemma 2.4

2.35
By Lemma 2.4 together with the approximation of the identity, we have lim

2.36
Then we complete the proof of this theorem.
-16 .And the inversion formula of the Radon transform by using inverse wavelet transform on H d was established in 17 .Our goal of the present article is to study the Calder ón reproducing formula on the Heisenberg group in L p -space with 1 < p < ∞.In the sequel we will develop the theory of inverse Radon transform on H d .Let L p H d be the space of measurable functions f on H d , such that We are now in a position to show that f ε,η converges to f in L 2 -space when ε → 0 and η → ∞.The result in this paper is an extension of that of Mourou and Trimèche 19 .Suppose that φ ∈ AW σ α and φ ∈ H σ α satisfies φ λ ∈ S ∞ H . Let Φ ε,η be defined by 2.10 .Then one has Φ ε,η ∈ L 2 H d .
Let k be a radial function inL p H d 1 < p < ∞ and let K z, t be defined by k in 2.27 .If k satisfies H d k z, t dz dt 0, and k z, t ln | z, t | ∈ L 1 H d , then K ∈ L 1 H d .∈H d and 0 < | z, t | < | z ,t |; by 2.25 we have * ≥ |K| a.e. on H d .
Mk is the Hardy-Littlewood maximal function of k.From the definition of K * , we haveK * ≤ 2 2d 2 Mk .Because k ∈ L p H d , we get K * ∈ L p H d .By the hypothesis k is radial and H d k z, t dz dt 0; together with the definition of K, Since k z, t ln | z, t | ∈ L 1 H d , it follows from Lemma 2.3 that H d \B K * z, t dz dt < ∞, that is, K * ∈ L 1 loc H d .On the other hand, K * ∈ L p H d ⊂ L 1 loc H d , and thus K and let K be defined by 2.27 .Suppose φ * φ k andH d K z, t dz dt 1.Then for f ∈ L p H d , one has lim ε → 0,η → ∞ f ε,η f.