Spherical means on M\'{e}tivier groups and support theorem

TL;DR

This paper characterizes functions with vanishing spherical means on Me9tivier groups, establishing support theorems and injectivity results for certain sets, advancing harmonic analysis in complex and non-commutative settings.

Contribution

It provides a characterization of spherical harmonic coefficients for functions with vanishing twisted spherical means and proves support and injectivity theorems in this context.

Findings

Characterization of spherical harmonic coefficients in terms of polynomial growth.

Support theorem for functions with vanishing twisted spherical means.

Sets of injectivity including non-harmonic complex cones and domain boundaries.

Abstract

Let $Z_{r, R}$ be the space of continuous functions on the annulus $B_{r, R}$ in $\mathbb C^n$ whose $\lambda$-twisted spherical mean, in the set up of the M\'{e}tivier group, vanishes over the spheres $S_s(z)\subset B_{r, R} $ with ball $B_r(0)\subseteq B_s(z).$ We characterize the spherical harmonic coefficients of functions in $Z_{r, R},$ eventually, in terms of polynomial growth, by which we infer support theorem. Further, we prove that non-harmonic complex cone and the boundary of a bounded domain are sets of injectivity for the $\lambda$-twisted spherical means.