Injectivity sets for spherical means on the Heisenberg group

Abstract

We prove that the boundary of a bounded domain is a set of injectivity for the twisted spherical means on C n for a certain class of functions on C n. As a consequence we obtain results about injectivity of the spherical mean operator in the Heisenberg group and the complex Radon transform.

A spectral Paley-Wiener theorem for the Heisenberg group and a support theorem for the twisted spherical means on \mathbb{C}^n

“…These type of rapid decay conditions are natural for the study of twisted spherical means as can be seen from the earlier works (see [1] [6] and [9]). …”

“…-As g is radial we have N g = 0 where N is the rotation operator (see(2.2)). Now it is well known that the equation (see [1] for eg., p. 365-366)…”

Twisted spherical means in annular regions in \mathbb{C}^n and support theorems

“…In the above theorem, the function f is assumed to have exponential decay, which reflects the non-Euclidean nature of the twisted spherical means. Such decay conditions also arise naturally in the integral geometry on the Heisenberg group as can be seen in the results in [1], [4]. However, the differentiability conditions on the function are genuine and cannot be relaxed.…”

Injectivity Sets for Spherical Means on the Heisenberg Group

“…Since ϕ k are Schwartz functions it follows that spheres are not sets of injectivity for the twisted spherical means on any L p n . Injectivity sets for the twisted spherical means have recently been studied by Agranovsky and Rawat [4]. For > 0 let V p be the class of functions f such that f z e 1/4+ z 2 is in L p n .…”

“…We say that a subset of n is a set of injectivity for the spherical mean value operator in V if f * σ r x = 0 for all r > 0 and x ∈ implies f = 0 for every f ∈ V . Determining sets of injectivity for the spherical means in various classes of functions is an interesting and important problem which has received considerable attention in recent times (see [2][3][4]). …”