Annihilating sets for the short time Fourier transform

We obtain a class of subsets of R2d such that the support of the short time Fourier transform (STFT) of a signal f∈L²(Rd) with respect to a window g∈L²(Rd) cannot belong to this class unless f or g is identically zero. Moreover we prove that the L²-norm of the STFT is essentially concentrated in the complement of such a set. A generalization to other Hilbert spaces of functions or distributions is also provided. To this aim we obtain some results on compactness of localization operators acting on weighted modulation Hilbert spaces.