The injectivity of the Pompeiu transform and L<SUP>p</SUP>-analogues of the Wiener-Tauberian theorem

Let E be a bounded Borel subset of Rn, n≥2, of positive Lebesgue measure and PE the corresponding 'Pompeiu transform'. We prove that PE is injective on Lp(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 Lq(Rn) for various values of q. We also take up the question of when PE 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