2 ILHAM A. ALIEV, BORIS RUBIN, SINEM SEZER, AND SIMTEN B. UYHAN

where

w

is an integrable radial function satisfying

fJRn

w(x)dx

=

0, have proved to

be a powerful tool in analysis and applications. There is a vast literature on this

subject (see, e.g.,

[Da], [HO], [M],

just for few). Owing to the formula

(1.1)

roo

ill - a~

Jo W f(x, t) tHa - Ca,w(

-~)

f(x),

aEC,

that can be given precise meaning, continuous wavelet transforms enable us to

resolve a variety of problems dealing with powers of differential operators. Such

problems arise, e.g., in potential theory, fractional calculus, and integral geometry;

see,

[HO], [Rl]-[R7], [Tr].

Dealing with functions of several variables, it is always

tempting to reduce the dimension of the domain of the wavelet function w and find

new tools to gain extra flexibility. This is actually a motivation for our article.

We introduce a new concept of the so-called composite wavelet transforms.

Loosely speaking, this is a class of wavelet-like transforms generated by two com-

ponents, namely, a kernel function and a wavelet. Both are in our disposal. The

first one depends on as many variables as we need for our problem. The second

component, which is a wavelet function (or a measure), depends only on one vari-

able. Such transforms are usually associated with one-parametric semigroups, like

Poisson, Gauss-Weierstrass, or metaharmonic ones, and can be implemented to

obtain explicit inversion formulas for diverse operators of the potential type and

fractional integrals. These arise in integral geometry in a canonical way; see, e.g.,

[H, R2, R6, R9].

In the present article we study different types of composite wavelet transforms

in the framework of the Lp-theory and the relevant Fourier and Fourier-Bessel

harmonic analysis. The main focus is reproducing formulas of Calderon's type

and explicit inversion of Riesz, Bessel, Flett, parabolic, and some other potentials.

Apart of a brief review of recent developments in the area, the paper contains

a series of new results. These include wavelet transforms for dilated kernels and

wavelet transforms generated by Beta-semigroups associated to multiplication by

exp(

-tl~li3),

f3

0, in terms of the Fourier transform. Such semigroups arise in the

context of stable random processes in probability and enjoy a number of remarkable

properties

[Ko], [La].

Special emphasis is made on detailed discussion of open

problems arising in wavelet analysis of functions of matrix argument. Important

results for £ 2-functions in this "higher-rank" set-up were obtained in

[OOR]

using

the Fourier transform technique. The

Lp-case

for p

=I

2 is still mysterious. The

main difficulties are related to correct definition and handling of admissible wavelet

functions on the cone of positive definite symmetric matrices.

The paper is organized according to the Contents presented above.

2. Composite Wavelet Transforms for Dilated Kernels

2.1. Preliminaries.

Let

Lp

=

Lp(lR.n),

1

~

p oo, be the standard space of

functions with the norm

II/IlP

=

(1n

lf(xW dx

f

1

p

oo.

For technical reasons, the notation

£

00

will be used for the space

C0

=

C0 (JR.n)

of

all continuous functions on

JR.n

vanishing at infinity. The Fourier transform of a