Coxeter system of planes are sets of injectivity for the twisted spherical means on $\mathbb C^n$

TL;DR

This paper proves that Coxeter systems of planes serve as sets of injectivity for twisted spherical means on complex spaces, extending previous results to more complex geometric configurations and function classes.

Contribution

It demonstrates that Coxeter systems of planes are sets of injectivity for twisted spherical means on complex spaces, generalizing from pairs of planes to systems with even numbers of planes.

Findings

Perpendicular planes are injectivity sets for TSM in $L^p(\mathbb C^n)$.

Coxeter systems of even number of planes are injectivity sets for TSM.

Certain product sets like $S_R^{2n-1} imes\mathbb C$ are injectivity sets for specific functions.

Abstract

In this article, we prove that any pair of perpendicular planes is a set of injectivity for the twisted spherical means (TSM) for $L^p(\mathbb C^n) (n\geq2)$ with $1\leq p\leq2.$ Then, we imitate that any Coxeter system of even number of planes is a set of injectivity for the TSM for $L^p(\mathbb C^n).$ We further observe that a set $S_R^{2n-1}\times\mathbb C$ is a set of injectivity for the TSM for a ceratin class of functions on $\mathbb C^{n+1}.$