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 $\Omega$ 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 $j_{q}: dom(\overline {\partial })\cap dom(\overline {\partial }^{*}) \hookrightarrow L^{2}_{(0,q)}(\Omega )$ is continuous in $W^{s}(\Omega )$-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.