Injectivity Sets for Spherical Means on the Heisenberg Group

Abstract

In this paper we prove that cylinders of the form R = S R × , where S R is the sphere z ∈ n z = R , are injectivity sets for the spherical mean value operator on the Heisenberg group H n in L p spaces. We prove this result as a consequence of a uniqueness theorem for the heat equation associated to the sub-Laplacian. A Hecke-Bochner type identity for the Weyl transform proved by D. Geller and spherical harmonic expansions are the main tools used. 

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].…”

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

“…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].…”

Revisiting Hardy's theorem for the Heisenberg group

“…First observe that when f = gP where g is radial and P ∈ H pq , f * λ ϕ λ k (z) has a simple form. Indeed, a simple calculation shows that (see [11])…”