Norm Estimates for Toeplitz Operators on the Bergman Space with Symbols Supported in Circles and Mixed Norms

Abstract

In this paper we study the class of Toeplitz operators $T_{\varphi }$, for which $\int _{0}^{1}\left( 1-t\right) ^{q-1}\left\| T_{\varphi d\sigma _{t}}\right\| _{p}^{q}dt<\infty ,$ for $1\le p,q<\infty ,$ where $\varphi \in L^{1}(\mathbb {D}),$ $\sigma _{t}$ is the Lebesgue measure in the circle $|\xi |=t$ and $\left\| \cdot \right\| _{p}$ is the p-Schatten norm of operators defined on the Bergman space of the disc. For that purpose we study the dependence on t of the norm and the p-Schatten norms of Toeplitz operators whose symbols are measures $\mu$ supported in the circle $tS^{1}$ with a positive density in $L^{1}(tS^{1})$.