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

Authors: AYŞE SANDIKÇI, AHMET TURAN GÜRKANLI

Abstract: 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 d\mu) (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.

Keywords: Sobolev-Shubin space, Gabor transform, modulation space, weighted Lorentz space, Toeplitz operators, Schatten-class

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