Sets of Injectivity for Weighted Twisted Spherical Means and Support Theorems

Abstract

Abstract. We prove that the spheres centered at origin are sets of injectivity for certain weighted twisted spherical means on C n . We also prove an analogue of Helgason's support theorem for weighted Euclidean and twisted spherical means.

Non-harmonic cones are sets of injectivity for the twisted spherical means on ℂⁿ

“…This real cone is not contained in the zero set of any bi-graded homogeneous harmonic polynomial. We would like to mention that the later result is a consequence of a result that R ∪ iR is set of injectivity for the TSM for L p (C), which has been proved by the author in the article [18].…”

“…The results on set of injectivity differ in the choice of sets and the class of functions considered. We would like to refer to [8,14,18], for some results on the sets of injectivity for the TSM.…”

“…For example, boundary of bounded domain and (R ∪ iR) × C n−1 are sets of injectivity for the TSM in C n having topological dimension 2n − 1. For more details see, [8,14,18].…”

“…They have shown that the boundary of any bounded domain in C n is set of injectvity for the TSM for a certain class of functions in L p (C n ). For more histories and further work on this question, we refer [5,6,14,18,19,20,21,25,26,27,28,29].…”

Real analytic expansion of spectral projections and extension of the Hecke-Bochner identity

“…Since this article is more concerned about Weyl correspondence and its applications, therefore, we skip here to write more histories of sets of injectivity for the twisted spherical means. We would like to refer [1,13,16,17,15].…”