Growth and Integrability of Fourier Transforms on Euclidean Space

摘要

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 -multipliers, a rather general inequality controlling the size of Fourier transforms for large and small argument is obtained. 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.