Spectral Analysis on Damek-Ricci Space

TL;DR

This paper develops spectral projection operators for Damek-Ricci spaces, establishing Paley-Wiener-Schwartz theorems, characterizing spectral ranges, and providing L^2 estimates, advancing harmonic analysis on these spaces.

Contribution

It introduces a spectral projection operator for Damek-Ricci spaces, characterizes its properties, and extends Paley-Wiener theorems beyond radial functions.

Findings

Spectral projection operator defined and studied on Damek-Ricci space.

Paley-Wiener-Schwartz theorem established for these spaces.

L^2-estimation for the spectral projection operator provided.

Abstract

We define and study the spectral projection operator for compactly supported distributions on Damek-Ricci space NA. The Paley-Wiener-Schwartz theorem and the range of S^{p}(NA)^{#}(0<p<=2) via spectral projection operator are established. The L^{2}-estimation for this operator is also given. In order to do the Paley-Wiener theorem for the non necessary radial function, the spectral projection operator can be uniquely characterized by analyticity and growth condition in lambda of Paley-Wiener theorem type on the unit disk of the complex plane as an example of Damek-Ricci space.