THE INJECTIVITY OF THE POMPEIU TRANSFORM AND L(P)-ANALOGS OF THE WIENER-TUBERIAN THEOREM

Let E be a bounded Borel subset of R(n), n greater than or equal to 2, of positive Lebesgue measure and P-E the corresponding 'Pompeiu trans form'. We prove that P-E is injective on L(p)(R(n)) if 1 less than or equal to p less than or equal to 2n/(n - 1). We explore the connection between this problem and a Wiener-Tauberian type theorem for the M(n) action on L(q)(R(n)) for various values of g. We also take up the que stion of when P-E is injective in case E is of finite, positive measur e, but is not necessarily a bounded set. Finally, we briefly look at t hese questions in the contexts of symmetric spaces of compact and non- compact type.