Gabor analysis of the spaces M(p,q,w) (ℝd) AND S(p,q,r,w,ω) (ℝd)

Let g be a non-zero rapidly decreasing function and w be a weight function. In this article in analog to modulation space, we define the space M(p,q,w) (ℝd) to be the subspace of tempered distributions f ɛ S'(ℝd) such that the Gabor transform Vg(f) of f is in the weighted Lorentz space L(p,q,wdμ)(ℝ2d). We endow this space with a suitable norm and show that it becomes a Banach space and invariant under time frequence shifts for 1≤p,q≤∞. We also investigate the embeddings between these spaces and the dual space of M(p,q,w)(ℝd). Later we define the space S(p,q,r,w,ω)ℝd for 1 < p < ∞, 1 ≤ q ≤ ∞. We endow it with a sum norm and show that it becomes a Banach convolution algebra. We also discuss some properties of S(p,q,r,w,ω)(ℝd). At the end of this article, we characterize the multipliers of the spaces M(p,q,w)(ℝd) and S(p,q,r,w,ω)(ℝd).