Sobolev regularity of the Bergman and Szegö projections in terms of $\overline{\partial}\oplus\overline{\partial}^{*}$ and $\overline{\partial}_{b}\oplus\overline{\partial}_{b}^{*}$

Abstract

Let $Ω$ be a smooth bounded pseudoconvex domain in $\mathbb{C}^{n}$. It is shown that for $0\leq q\leq n$, $s\geq 0$, the embedding $j_{q}: dom(\overline{\partial})\cap dom(\overline{\partial}^{*}) \hookrightarrow L^{2}_{(0,q)}(Ω)$ is continuous in $W^{s}(Ω)$--norms if and only if the Bergman projection $P_{q}$ is (see below for the modification needed for $j_{0}$). The analogous result for the operators on the boundary is also proved (for $n\geq 3$). In particular, $j_{1}$ is always regular in Sobolev norms in $\mathbb{C}^{2}$, notwithstanding the fact that $N_{1}$ need not be.

Theorems & Definitions (2)

• Theorem 1.1
• proof : Proof of Theorem \ref{['equivalent']}