The injectivity of the Pompeiu transform and L^p-analogues of the Wiener-Tauberian theorem

Abstract

Let E be a bounded Borel subset of Rⁿ, n≥2, of positive Lebesgue measure and P_E the corresponding 'Pompeiu transform'. We prove that P_E is injective on L^p(Rⁿ) if 1≤p≤2n/(n−1). We explore the connection between this problem and a Wiener-Tauberian type theorem for theM(n) action on L^q(Rⁿ) for various values of q. We also take up the question of when P_E is injective in caseE is of finite, positive measure, but is not necessarily a bounded set. Finally, we briefly look at these questions in the contexts of symmetric spaces of compact and non-compact type.