Carleson type measures on parabolic Bergman spaces
The \(\alpha\)-parabolic Bergman space \(b_\alpha^p\) is the space of all \(p\)-th integrable solutions \(u\) of the parabolic equation \([\partial/\partial t+(-\Delta)^\alpha]u=0\) on the upper half-space, where \(0<\alpha\leq 1\) and \(1\leq p<\infty\). In case \(\alpha=1/2\), the space \(b_{1/2}^p\) coincides with the well-known harmonic Bergman space. The main result of this paper is the characterization of Carleson measures for the derivatives of \(b^p_\alpha\)-functions. In order to obtain such a result, the authors first establish the upper and lower estimates for the derivatives of the fundamental solution. As an application, the authors obtain a characterization of positive symbol functions of bounded Toeplitz operators on \(b^2_\alpha\) in terms of averaging functions and Berezin transforms. In case \(\alpha=1/2\), some of these results and more were known by \textit{B. R. Choe, H. Koo} and \textit{H. Yi} [Potential Anal. 17, 307--335 (2002; Zbl 1014.47014)]
scientific article; zbMATH DE number 5015990