Generalized Sobolev-Shubin spaces, boundedness and Schatten class properties of Toeplitz operators

Abstract

Let w and omega be two weight functions on R-2d and 1 <= p, q <= infinity. Also let M (p, q, omega) (R-d) denote the subspace of tempered distributions S' (R-d) consisting of f is an element of S' (R-d) such that the Gabor transform V(g)f of f is in the weighted Lorentz space L (p, q, wd mu) (R-2d). In the present paper we define a space Q(g,omega)(M(p,q,omega)) (R-d) as counter image of M (p, q, omega) (R-d) under Toeplitz operator with symbol omega. We show that Q(g,omega)(M(p,q,omega)) (R-d) is a generalization of usual Sobolev-Shubin space Q(s) (R-d). We also investigate the boundedness and Schatten-class properties of Toeplitz operators.