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

TL;DR

This paper investigates the Sobolev regularity of the Bergman and Szeg"o projections on smooth bounded pseudoconvex domains in complex spaces, establishing conditions for their boundedness in Sobolev spaces related to the $ar{ullstop}$ and boundary $ar{ullstop}_b$ operators.

Contribution

It provides a characterization of Sobolev regularity for the Bergman and Szeg"o projections in terms of the $ar{ullstop} ext{-}ar{ullstop}^*$ and boundary operators, extending previous results to boundary cases.

Findings

Embedding $j_q$ is continuous in $W^s$ if and only if the Bergman projection $P_q$ is regular.

On the boundary, similar regularity results hold for $n eq 2$.

In $C^2$, $j_1$ is always Sobolev regular regardless of the regularity of $N_1$.

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.