  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.
