BMO, VMO and Hankel operators on the Bergman space of strongly pseudoconvex domains
For a bounded pseudoconvex domain \(D\) with smooth boundary in \(\mathbb{C}^ n\), the author considers Hankel operators \(H_ f\) acting on the Bergmann space \(H^ 2(D)\), consisting of the holomorphic \(L^ 2\) functions on \(D\). For \(f\in L^ 2(D)\), he proves that \(H_ f\) and \(H_{\bar f}\) are bounded provided that \(f\) has ``bounded mean oscillation'' on \(D\) (in the sense that he defines), and that \(H_ f\), \(H_{\bar f}\) are compact if \(f\) has ``vanishing mean oscillation at the boundary of \(D\)''. For \(f\in H^ 2(D)\), these conditions are also necessary. Analogous results for bounded symmetric domains were proved by \textit{D. Békollé}, \textit{C. A. Berger}, \textit{L. A. Coburn} and \textit{K. H. Zhu} [J. Funct. Anal. 93, 310-350 (1990; Zbl 0765.32005)], and the present results partially answer a conjecture posed at the end of that paper.
vanishing mean oscillation at the boundary
scientific article; zbMATH DE number 40704
scientific article