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

TL;DR

This paper investigates the smoothing effects of the Bergman projection on holomorphic functions in complex domains, revealing directional derivative control and automatic regularity of conjugate holomorphic functions.

Contribution

It establishes new smoothing results for the Bergman projection, linking Sobolev norms to directional derivatives and demonstrating automatic regularity of conjugate holomorphic functions.

Findings

Full Sobolev norm controlled by directional derivatives

Projection of conjugate holomorphic functions is smoother

Corollaries for global regularity 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 $\|Bf\|_{k_{2}}$ is controlled by $L^2$ derivatives of $f$ taken along a single, distinguished direction (of order $\leq k_{1}$), 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.