L ^p-Integrability, Supports of Fourier Transforms and Uniqueness for Convolution Equations

Abstract

It is proved that a nonzero function is not in L ^p(R ⁿ) with p \le 2n/d if its Fourier transform is supported by a d-dimensional submanifold. It is shown that the assertion fails for p > 2n/d and d \ge n/2. The result is applied for obtaining uniqueness theorems for convolution equations in L ^p-spaces.