Sharp Embeddings for Modulation Spaces and the Poisson Summation Formula

In this paper we sharpen a theorem of Grochenig about embedding weighted L p -spaces into a modulation space. The theorem is closely related to the problem of pointwise convergence in the Poisson summation formula (PSF). We also show that this sharp version is optimal in some sense by extending a family of counterexamples to PSF also due to Grochenig. The proof requires new techniques, namely the interplay between modulation spaces and Lorentz spaces. We also establish embeddings with weighted L p -norms replaced by weighted amalgam space norms. 1. MODULATION SPACES The modulation space M w p;q (R n ) consists of functions f whose short-time Fourier transforms (STFTs) belong to a certain mixed-norm space on R n \Theta R n . Here, the STFT is the bilinear mapping S(f; g)(t; !) = Z f(v)e \Gamma2ßi! \Deltav g(t \Gamma v) dv j D f; M! g T t g E ; (t; !) 2 R n \Theta R n where M! denotes the modulation operator of multiplication by e 2ßi! \Deltav , T t denotes tr..