Sharp continuity results for the short-time Fourier transform and for localization operators

We completely characterize the boundedness on Wiener amalgam spaces of the short-time Fourier transform (STFT), and on both L^p and Wiener amalgam spaces of a special class of pseudodifferential operators, called localization operators. Precisely, sufficient conditions for the STFT to be bounded on the Wiener amalgam spaces W(L^p , L^q ) are given and their sharpness is shown. Localization operators are treated similarly: using different techniques from those employed in the literature, we relax the known sufficient boundedness conditions for these operators to be bounded on L^p spaces and prove the optimality of our results. Next, we exhibit sufficient and necessary conditions for such operators to be bounded on Wiener amalgam spaces.