Schatten class Toeplitz operators on the parabolic Bergman space. II. (English) Zbl 1243.35076

Summary: Let \(0 < \alpha \leq 1\) and let \(b^2_{\alpha}\) be a Hilbert space of all square integrable solutions of a parabolic equation \((\partial _{t} + ( - \Delta )^{\alpha })u = 0\) on the upper half space. We study the Toeplitz operators on \(b^2_{\alpha}\), 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.
For Part I see [the authors, Kodai Math. J. 32, No. 3, 501–520 (2009; Zbl 1198.47046)].


35K05 Heat equation
31B10 Integral representations, integral operators, integral equations methods in higher dimensions
26D10 Inequalities involving derivatives and differential and integral operators


Zbl 1198.47046
Full Text: DOI Euclid