Injectivity sets for spherical means on the Heisenberg group

Abstract

In this paper we prove that cylinders of the form Γ_R = S_R × R, where S_R is the sphere {z ∈ Cⁿ: |z| = R}, are injectivity sets for the spherical mean value operator on the Heisenberg group Hⁿ 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.