Injectivity of spherical means on H-type groups

TL;DR

This paper proves that certain spherical averaging operators on H-type groups are injective for functions in specific L^p spaces, extending understanding of harmonic analysis on these groups.

Contribution

It establishes injectivity of three types of spherical means on H-type groups for a range of p, with sharpness results for the first two cases.

Findings

Injectivity holds for 1 ≤ p ≤ 2m/(m-1) in all three cases.

Sharpness of results demonstrated with examples for the first two cases.

Results extend harmonic analysis tools on H-type groups.

Abstract

We establish injectivity results for three different spherical means on an $H$-type group, $G$. First is the standard spherical means which is defined to be the average of a function over the spheres in the complement of the center, second is the average over the product of spheres in the center and its complement, and the third is the average over the spheres defined by a homogeneous norm on $G$. If $m$ is the dimension of the center of $G$, injectivity of these spherical means is proved for the range $1 \leq p \leq \frac{2m}{m-1}$. Examples are provided to show the sharpness of our results in the first two cases.