Support theorem on R^n and non compact symmetric spaces

TL;DR

This paper establishes conditions under which solutions to certain convolution equations imply the compact support of functions in both Euclidean and non-compact symmetric spaces, extending classical support theorems.

Contribution

It generalizes support theorems for convolution equations to non-compact symmetric spaces, under specific conditions on the Fourier transform's zero set.

Findings

If g is compactly supported, then f is also compactly supported under given conditions.

Results apply to convolution equations on R^n and non-compact symmetric spaces.

Provides a framework for understanding support properties in more general geometric contexts.

Abstract

We consider convolution equations of the type f * T = g where f, g are in L^p(R^n) and T is a compactly supported distribution. Under natural assumptions on the zero set of the Fourier transform of T we show that f is compactly supported, provided g is. Similar results are proved for non compact symmetric spaces as well.