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

Abstract

In this article, we prove that any pair of perpendicular planes is a set of injectivity for the twisted spherical means (TSM) for $L^p({\mathbb{C}}^n)(n\ge 2)$ with $1\le p\le 2.$ 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 ${\mathbb{C}}^{n+1}.$

arXiv commentThis article is accepted in J. Funct. Anal. jointly with arXiv:1103.4571