Growth and integrability of Fourier transforms on Euclidean space

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.