Hankel operators on weighted Bergman spaces on strongly pseudoconvex domains (Q1328082)

The paper concerns the study of the Hankel operator \(H_ f\) and the non- orthogonal Hankel operator \(\widetilde H_ f\); i.e. \[ H_ f g(z)= (I- P)(\widetilde f g)(z);\;\widetilde H_ f g(z)= (I- \widetilde P)(\widetilde f g)(z), \] where \(P(\widetilde P)\) is the (non-)orthogonal projection \(L^ 2\nu\to A^{2,\nu}\) (\(A^{2,\nu}\) the weighted Bergman space = the closed subspace in \(L^ 2_ \nu\) of holomorphic functions; \(L^ 2_ \nu= L^ 2\)-space \(L^ 2(\Omega,dm_ \nu)\) on a \(C^ \infty\)-bounded strongly pseudoconvex domain \(\Omega\subset \mathbb{C}^ n\) and \(dm_ \nu= |\rho(z)|^ \nu dm\), \(dm=\) Lebesgue volume form). The main results of the paper are Theorem 1. Let \(f\in {\mathcal H}(\Omega)\). Then the following are equivalent. i) \(f\in {\mathcal B}\); ii) \(H_ f\) is bounded; iii) \(\widetilde H_ f\) is bounded. Theorem 2. Let \(f\in {\mathcal H} (\Omega)\). Then the following are equivalent. i) \(f\in {\mathcal B}_ 0\); ii) \(H_ f\) is compact; iii) \(\widetilde H_ f\) is compact. (Here \({\mathcal H}(\Omega)=\) holomorphic functions space on \(\Omega\); \[ {\mathcal B}= \{f| f\in {\mathcal H}(\Omega),\;\sup_{z\in \Omega} | \widetilde D f(z)|< \infty\} \] with \(\widetilde D\) the covariant derivative, and \(f\in {\mathcal B}_ 0\) if \(f\in {\mathcal B}\) with \(\lim_{z\in \partial\Omega} |\widetilde Df(z)|= 0\)).