Research Article | Open Access

Jinsen Xiao, Jianxun He, "The Calderón Reproducing Formula Associated with the Heisenberg Group ", *Mathematical Problems in Engineering*, vol. 2010, Article ID 132186, 12 pages, 2010. https://doi.org/10.1155/2010/132186

# The Calderón Reproducing Formula Associated with the Heisenberg Group

**Academic Editor:**Victoria Vampa

#### Abstract

We establish the Calderón reproducing formula for functions in on the Heisenberg group . Also, we develop this formula in with .

#### 1. Introduction

The classical Calderón reproducing formula reads where , and denotes the convolution on 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 when and are sufficiently nice normalized radial wavelet functions. The generalization of (1.1) involving nonradial wavelets and can be written in the following form: where and are rotated versions of and on . The authors in [8, 9] established (1.1) for Holschneider [10] studied the formula (1.2) in case and gave an inversion formula of the Radon transform in -space by using wavelets. Furthermore, Rubin [4] developed the Calderón reproducing formula, windowed X-ray transforms, the Radon transforms, and -plane transforms in -spaces on .

It is a remarkable fact that the Heisenberg group, denoted by , 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 in [11–13] and the references therein. Also, the researches of wavelet analysis on 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 was established in [17]. Our goal of the present article is to study the Calderón reproducing formula on the Heisenberg group in -space with . In the sequel we will develop the theory of inverse Radon transform on .

The Heisenberg group is a Lie group with the underlying manifold , and the multiplication law is given by where The dilation of is defined by with For , the homogeneous norm of is given by (see [11]) Notice that In addition, satisfies the quasitriangle inequality: The homogeneous dimension of is and the volume of a ball is , where is a constant.

Let ; then is a locally compact nonunimodular group with the group law The left and right Haar measures on are given by where denotes the Lebesgue measure on

Let be the space of measurable functions on , such that

Let and The Fock space is the space of holomorphic functions on such that From [18] we know that is an orthonormal basis of the Hilbert space For let be the Bargmann-Fock representation of which acts on by The group Fourier transform of a function is defined by

Let be the classes of Schatten-von Neumann operators on Hilbert space and let denote the algebra of all bounded operators, that is, For let denote the -norm of If is just the Hilbert-Schmit norm of that denotes Let denote the usual operator norm of in For let be the Banach space consisting of all weak measurable operator value functions , which also satisfy a.e. and For the Parseval formula is where denotes the adjoint of The Plancherel formula is As a consequence of (1.13), one has the inversion of the Fourier transform: Suppose and let By a direct computation, we have Let be the convolution of and that is, Then We should notice the following facts: if then And if , then The further detail of harmonic analysis on 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 on is defined by Let denote the set of all positive real numbers, Let be the projection from to -dimensional subspace spanned by and let From [15, Theorem ], we have

Let and , ; if and satisfies then we call an admissible wavelet and write Let ; the continuous wavelet transform of with respect to is defined by And the following Calderón reproducing formula holds in the weak sense:

##### 2.1. Calderón Reproducing Formula in

By (1.16) and (1.18), we can rewrite (2.5) and (2.6) as follows:

For let then We are now in a position to show that converges to in -space when and The result in this paper is an extension of that of Mourou and Trimèche [19].

Lemma 2.1. *Suppose that and satisfies Let be defined by (2.10). Then one has *

*Proof. *By Hölder's inequality, we have
Thus
By (1.14), (1.19), and (1.20), we have
Noticing that
we obtain
Thus we get
Therefore,
Then we complete the proof of this lemma.

Theorem 2.2. *Let and satisfy Then for one has and *

*Proof. *Notice that
and by (1.19) and Lemma 2.1 we deduce Then by (1.17) and (1.19), we have
where if otherwise By (1.11) we get the desired result.

##### 2.2. Calderón Reproducing Formula in with

For with , the continuous wavelet transform of with respect to a wavelet can be defined by formula (2.7) under certain conditions on . In this part we will show that converges to

Let be a measurable function on ; for define It is easy to see that is nonnegative and radially decreasing, that is, and a.e. on

Let Then for any , we define and thus we have the following lemma.

Lemma 2.3. *Let and be defined by (2.22). Then one has
**
where is a positive constant, .*

*Proof. *First we let
It is obvious that for any Since is nonnegative and radially decreasing, from [11, page 542], we know that there exists a positive constant such that
Thus,
This completes the proof.

Let be a radial function in () and let for any , where is a unit vector of Then we have the following lemma.

Lemma 2.4. *Let be a radial function in and let be defined by in (2.27). If satisfies and then *

*Proof. *Analogous to (2.20) we define
and then a.e. on

Let and ; by (2.25) we have
where is the Hardy-Littlewood maximal function of . From the definition of we have Because we get

By the hypothesis is radial and ; together with the definition of we have
Since it follows from Lemma 2.3 that that is, On the other hand, and thus which implies that

Without loss of generality, we assume that ; then (2.9) and (2.10) can be written as

In fact, is always stated under conditions on rather than under conditions on for convenience (see [4, 10]). By Lemma 2.4 we have the following theorem.

Theorem 2.5. *Let be in the conditions of Lemma 2.4 and let be defined by (2.27). Suppose and Then for one has *

*Proof. *From (2.31) we have
Since and we deduce
Then we have
By Lemma 2.4 together with the approximation of the identity, we have
Then we complete the proof of this theorem.

#### Acknowledgments

The work for this paper is supported by the National Natural Science Foundation of China (no.s 10671041 and 10971039) and the Doctoral Program Foundation of the Ministry of Education of China (no. 200810780002).

#### References

- V. S. Guliyev and E. J. Ibrahimov, “Calderón reproducing formula associated with the Gegenbauer operator on the half-line,”
*Journal of Mathematical Analysis and Applications*, vol. 335, no. 2, pp. 1079–1094, 2007. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - Y. Han, S. Lu, and D. Yang, “Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type,”
*Journal of Fourier Analysis and Applications*, vol. 7, no. 6, pp. 571–600, 2001. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Holschneider and P. Tchamitchian, “Pointwise analysis of Riemann's “nondifferentiable” function,”
*Inventiones Mathematicae*, vol. 105, no. 1, pp. 157–175, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Rubin, “The Calderón reproducing formula, windowed X-ray transforms, and Radon transforms in ${L}^{p}$-spaces,”
*Journal of Fourier Analysis and Applications*, vol. 4, no. 2, pp. 175–197, 1998. View at: Publisher Site | Google Scholar | MathSciNet - I. Daubechies,
*Ten Lectures on Wavelets*, vol. 61 of*CBMS-NSF Regional Conference Series in Applied Mathematics*, SIAM, Philadelphia, Pa, USA, 1992. View at: MathSciNet - Y. Meyer,
*Wavelets and Operators*, vol. 37 of*Cambridge Studies in Advanced Mathematics*, Cambridge University Press, Cambridge, UK, 1992. View at: MathSciNet - M. Frazier, B. Jawerth, and G. Weiss,
*Littlewood-Paley Theory and the Study of Function Spaces*, vol. 79 of*CBMS Regional Conference Series in Mathematics*, American Mathematical Society, Providence, RI, USA, 1991. View at: MathSciNet - M. A. Mourou and K. Trimèche, “Calderon's reproducing formula related to the Dunkl operator on the real line,”
*Monatshefte für Mathematik*, vol. 136, no. 1, pp. 47–65, 2002. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - B. Rubin and E. Shamir, “Calderón's reproducing formula and singular integral operators on a real line,”
*Integral Equations and Operator Theory*, vol. 21, no. 1, pp. 77–92, 1995. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - M. Holschneider, “Inverse Radon transforms through inverse wavelet transforms,”
*Inverse Problems*, vol. 7, no. 6, pp. 853–861, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - E. M. Stein,
*Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals*, vol. 43 of*Princeton Mathematical Series*, Princeton University Press, Princeton, NJ, USA, 1993. View at: MathSciNet - D. Geller, “Fourier analysis on the Heisenberg group. I. Schwartz space,”
*Journal of Functional Analysis*, vol. 36, no. 2, pp. 205–254, 1980. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - R. S. Strichartz, “${L}^{p}$ harmonic analysis and Radon transforms on the Heisenberg group,”
*Journal of Functional Analysis*, vol. 96, no. 2, pp. 350–406, 1991. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - J. He and H. Liu, “Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane,”
*Journal of Mathematical Analysis and Applications*, vol. 208, no. 1, pp. 58–70, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Liu and L. Peng, “Admissible wavelets associated with the Heisenberg group,”
*Pacific Journal of Mathematics*, vol. 180, no. 1, pp. 101–123, 1997. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet - H. Liu, Y. Liu, and H. Wang, “Multiresolution analysis, self-similar tilings and Haar wavelets on the Heisenberg group,”
*Acta Mathematica Scientia B*, vol. 29, no. 5, pp. 1251–1266, 2009. View at: Publisher Site | Google Scholar | MathSciNet - J. He, “An inversion formula of the Radon transform on the Heisenberg group,”
*Canadian Mathematical Bulletin*, vol. 47, no. 3, pp. 389–397, 2004. View at: Google Scholar | Zentralblatt MATH | MathSciNet - J. He, “Continuous multiscale analysis on the Heisenberg group,”
*Bulletin of the Korean Mathematical Society*, vol. 38, no. 3, pp. 517–526, 2001. View at: Google Scholar | Zentralblatt MATH | MathSciNet - M. A. Mourou and K. Trimèche, “Calderón's formula associated with a differential operator on $(0,\infty )$ and inversion of the generalized Abel transform,”
*Journal of Fourier Analysis and Applications*, vol. 4, no. 2, pp. 229–245, 1998. View at: Publisher Site | Google Scholar | Zentralblatt MATH | MathSciNet

#### Copyright

Copyright © 2010 Jinsen Xiao and Jianxun He. 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.