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

Let w and omega be two weight functions on R^{2d} and 1 leq p,q leq infty. Also let M(p,q,omega) (R^d) denote the subspace of tempered distributions S (R^d) consisting of f in S (R^d) such that the Gabor transform V_g f of f is in the weighted Lorentz space L(p,q,omega dmu) (R^{2d}) . In the present paper we define a space Q_{g,w}^{M(p,q,omega) (R^d) as counter image of M(p,q,omega) (R^d) under Toeplitz operator with symbol w. We show that Q_{g,w}^{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.