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

TL;DR

This paper investigates the relationship between the $L^p$-integrability of functions, the support size of their Fourier transforms, and the implications for Wiener-Tauberian theorems, revealing critical thresholds for support measures.

Contribution

It establishes new bounds on $L^p$ functions with Fourier support on sets of finite packing measure, extending Wiener-Tauberian results to broader contexts.

Findings

No non-zero $L^p$ functions with Fourier support on finite packing measure sets for $1 \\leq p \\leq 2n/\alpha$

Failure of this property for $p > 2n/\alpha$

Application to $L^p$ Wiener-Tauberian theorems for $\mathbb{R}^n$ and $M(2)$

Abstract

It is proved that there does not exist any non zero function in $L^p(\R^n)$ with $1\leq p\leq 2n/\alpha$ 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 $\R^n$ and M(2).