Sets of injectivity for weighted twisted spherical means and support theorems

TL;DR

This paper establishes that spheres in complex space serve as sets of injectivity for weighted twisted spherical means with spherical harmonic weights, contributing to the understanding of support theorems in harmonic analysis.

Contribution

It proves that spheres are sets of injectivity for weighted twisted spherical means with spherical harmonic weights in complex space.

Findings

Spheres are sets of injectivity for WTSM with spherical harmonic weights.

Addresses the open problem for real analytic weights in twisted spherical means.

Provides a foundation for further support theorems in complex harmonic analysis.

Abstract

In this article, we show that the spheres $S_R(o)=\{z\in\mathbb C^n: |z|=R\}$ are sets of injectivity for the weighted twisted spherical means (WTSM) for a suitable class of functions on $\mathbb C^n$. The weights here are spherical harmonics on $S^{2n-1}.$ In general, the question of set of injectivity for the twisted spherical means (TSM) with real analytic weight is still open. We would like to refer to \cite{NRR}, for some results on the sets of injectivity for the spherical means with real analytic weights in the Euclidean setup.