Growth and integrability of Fourier transforms on Euclidean space

TL;DR

This paper explores the relationship between function smoothness and Fourier transform behavior, establishing new inequalities and integrability results that extend classical one-dimensional Fourier analysis to higher dimensions.

Contribution

It introduces a general inequality for Fourier transforms using $L^{p}$-multipliers, extending Titchmarsh's ideas to multiple dimensions and providing quantitative Riemann-Lebesgue estimates.

Findings

Established a general inequality controlling Fourier transform size

Derived quantitative Riemann-Lebesgue estimates

Extended integrability results of Fourier transforms to higher dimensions

Abstract

A fundamental theme in classical Fourier analysis relates smoothness properties of functions to the growth and/or integrability of their Fourier transform. By using a suitable class of $L^{p}-$multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is proved. As consequences, quantitative Riemann-Lebesgue estimates are obtained and an integrability result for the Fourier transform is developed extending ideas used by Titchmarsh in the one dimensional setting.