\(L^p\)-integrability, supports of Fourier transforms and uniqueness for convolution equations (Q1762630)

The main result of this paper relates the dimension of the support of the Fourier transform of a nonzero function with its membership in \(L^p({\mathbb R}^n)\). The authors prove that a nonzero function whose Fourier transform is supported by a \(C^1\) manifold of dimension \(d<n\) (\(d>0\)) is not in \(L^p({\mathbb R}^n)\) for \(1\leq p\leq 2n/d\). Moreover, they show that the estimate for index \(p\) is sharp when the dimension \(d\) verifies \(n/2\leq d\leq n\). These results are applied to study uniqueness theorems for convolution equations of the form \(f*T=0\), where \(f\) belongs to some \(L^p\) spaces and \(T\) is a distribution on \({\mathbb R}^n\) of compact support. As application it is also obtained a characterization of stationary sets of evolution equations. Thus the result on closed stationary hypersurfaces proved by \textit{M. Agranovsky}, \textit{C. Berenstein} and \textit{P. Kuchment} [J. Geom. Anal. 6, No. 3, 365--383 (1996; Zbl 0898.44003)] is generalized, from the wave equation to evolution equations and for more general di\-ffe\-ren\-tial operators than the Laplacian.