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A fundamental theme in classical Fourier analysis relates smoothnessproperties of functions to the growth and/or integrability of their Fouriertransform. By using a suitable class of $L^{p}-$multipliers, a rather generalinequality controlling the size of Fourier transforms for large and smallargument is proved. As consequences, quantitative Riemann-Lebesgue estimatesare obtained and an integrability result for the Fourier transform is developedextending ideas used by Titchmarsh in the one dimensional setting.

Growth and integrability of Fourier transforms on Euclidean space