Duality of holomorphic functions spaces und smoothing properties of the Bergman projection

Abstract

Let $\Omega \subset {\mathbb{C}}^n$ be a bounded domain with smooth boundary, whose Bergman projection $B$ maps the Sobolev space $H^{k_1}(\Omega )$ (continuously) into $H^{k_2}(\Omega )$. We establish two smoothing results: (i) the full Sobolev norm $\Vert Bf{\Vert }_{k_2}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\le k_1$), and (ii) the projection of a conjugate holomorphic function in $L^2(\Omega )$ is automatically in $H^{k_2}(\Omega )$. There are obvious corollaries for when $B$ is globally regular.

Cited by in corpus (1)

• A note on a smoothing property of the harmonic Bergman projection