Sharp Embeddings for Modulation Spaces and the Poisson Summation Formula

In this paper we sharpen a theorem of Gröchenig about embedding weighted Lp-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 Gröchenig. The proof requires new techniques, namely the interplay between modulation spaces and Lorentz spaces. We also establish embeddings with weighted Lp-norms replaced by weighted amalgam space norms. 1. MODULATION SPACES The modulation space Mw p;q(Rn) consists of functions f whose short-time Fourier transforms (STFTs) belong to a certain mixed-norm space on Rn Rn. Here, the STFT is the bilinear mapping S(f; g)(t; !) = Z f(v)e 2 i! vg(t v) dv Df;M!gTtgE ; (t; !) 2 Rn Rn where M! denotes the modulation operator of multiplication by e2 i! v, Tt denotes translation g(v) 7! g(v t), h 7! eh( ) = h( ) is the usual involution and h ; i extends the L2 inner product to dual function spaces on Rn. The identity kS(f; g)k22 = kfk22 kgk22 ; (1) (e.g., [3]) suggests fixing a ‘window’ g 2 L1 \ L1(Rn) with kgk22 = 1, say, and then regarding f 7! S(f; g) as an isometric embedding of L2(Rn) into L2(Rn Rn); one can define modulation spaces by using other measures of the size of S(f; g): DEFINITION 1.1. Given 1 p; q < 1 the modulation space Mpq(Rn) consists of those f such that Z Z jS(f; g)(t; !)jp dt q=p d! <1: We then denote by kfkMpq the q-th root of this quantity. When p = q we write Mpq = Mp. Obvious adjustments can be made for the cases where p and/or q are infinite. One can insert a nonnegative weight into the norm in Definition 1.1 as was done by Feichtinger and Gröchenig (cf. [3]). We also note that different window functions g give rise to equivalent norms on Mw pq(Rn). 2. MODULATION EMBEDDINGS AND POISSON SUMMATION