Singular Value Estimates for Certain Convolution-Product Operators

Abstract

This paper presents an expansion for radial tempered distributions on ${\bf R}^n$ in terms of smooth, radial analyzing and synthesizing functions with space-frequency localization properties similar to standard wavelets. Scales of quasi-norms are defined for the coefficients of the expansion that characterize, via Littlewood-Paley-Stein theory, when a radial distribution belongs to a Triebel-Lizorkin or Besov space. These spaces include, for example, the $L^p$ spaces, $1 < p < \infty,$ Hardy spaces $H^p, 0 < p \leq 1,$ Sobolev spaces $L^p_k,$ and Lipschitz spaces $\Lambda_\alpha, \alpha > 0.$ We also present a smooth radial atomic decomposition and norm estimates for sums of smooth radial molecules. The radial wavelets, atoms, and molecules that we consider are localized near certain annuli, as opposed to cubes in the usual, nonradial setting. The radial wavelet expansion is multiscale, where the functions in the different scales are related by dilation. However, there is no translation structure within a given scale, unlike the situation with standard wavelet systems.