Non-harmonic cones are sets of injectivity for the twisted spherical means on $\mathbb C^n$

TL;DR

This paper characterizes complex cones that serve as sets of injectivity for twisted spherical means on ^n, identifying conditions related to harmonic polynomials and providing explicit examples.

Contribution

It establishes a precise geometric criterion for injectivity sets in twisted spherical means and constructs explicit examples of such sets.

Findings

Complex cones not lying on harmonic polynomial level surfaces are injectivity sets.

Provides explicit examples of level surfaces that are injectivity sets.

Characterizes the geometric structure of injectivity sets for twisted spherical means.

Abstract

In this article, we prove that a complex cone is a set of injectivity for the twisted spherical means for the class of all continuous functions on $\mathbb C^n$ as long as it does not completely lay on the level surface of any bi-graded homogeneous harmonic polynomial on $\mathbb C^n.$ Further, we produce examples of such level surfaces.