An analogue of Bochner's theorem for Damek-Ricci spaces

TL;DR

This paper extends Bochner's theorem to Damek-Ricci spaces by characterizing the spherical transform of radial positive measures on harmonic $NA$ groups, linking measure properties to their transforms.

Contribution

It provides a new characterization of the spherical transform of radial positive measures on Damek-Ricci spaces, generalizing classical harmonic analysis results.

Findings

Characterization of the spherical transform of radial positive measures.

Extension of Bochner's theorem to harmonic $NA$ groups.

Conditions for measures to have finite integral against elementary spherical functions.

Abstract

We characterize the image of radial positive measures $\theta$'s on a harmonic $NA$ group $S$ which satisfies $\int_S\phi_0(x)\,d\theta(x)<\infty$ under the spherical transform, where $\phi_0$ is the elementary spherical function.