$L^p$-integrability, dimensions of supports of fourier transforms and applications

Abstract

It is proved that there does not exist any non zero function in $L^p({\mathbb{R}}^n)$ with $1\le p\le 2n/\alpha$ if its Fourier transform is supported by a set of finite packing $\alpha$-measure where $0<\alpha <n$. It is shown that the assertion fails for $p>2n/\alpha$. The result is applied to prove $L^p$ Wiener-Tauberian theorems for ${\mathbb{R}}^n$ and M(2).

References in corpus (1)

• Wiener's 'closure of translates' problem and Piatetski-Shapiro's uniqueness phenomenon