Convolution operators defined by singular measures on the motion group

TL;DR

This paper establishes an $L^{p}$ improving property for convolution operators linked to singular measures on the motion group, extending Radon transform results using harmonic analysis techniques.

Contribution

It introduces a new $L^{p}$ improving result for convolution operators on the motion group based on mild geometric conditions of hypersurfaces.

Findings

Proves $L^{p}$ improving for certain convolution operators

Extends Radon transform results to the motion group

Uses harmonic analysis on the motion group

Abstract

This paper contains an $L^{p}$ improving result for convolution operators defined by singular measures associated to hypersurfaces on the motion group. This needs only mild geometric properties of the surfaces, and it extends earlier results on Radon type transforms on $\mathbb{R}^{n}$. The proof relies on the harmonic analysis on the motion group.