Schatten class Toeplitz operators on the parabolic Bergman space II

Abstract

Let 0 < α ≤ 1 and let $\boldsymbol{b}_\alpha^{2}$ be a Hilbert space of all square integrable solutions of a parabolic equation (∂_t + (−Δ)^α)u = 0 on the upper half space. We study the Toeplitz operators on $\boldsymbol{b}_\alpha^{2}$, which we characterize to be of Schatten class whose exponent is smaller than 1. For the proof, we use an atomic decomposition theorem of parabolic Bergman functions. Generalizations to Schatten class operators for Orlicz type and Herz type are also discussed.