Sobolev regularity of the Bergman and Szegö projections in terms of $\bar{\mathrm{\partial }}\oplus {\bar{\mathrm{\partial }}}^{\ast }$ and ${\bar{\mathrm{\partial }}}_b\oplus {\bar{\mathrm{\partial }}}_b^{\ast }$

Let $\mathrm{\Omega }$ be a smooth bounded pseudoconvex domain in ${\mathbb{C}}^n$. It is shown that for $0\le q\le n$, $s\ge 0$, the embedding $j_q:dom(\bar{\mathrm{\partial }})\cap dom({\bar{\mathrm{\partial }}}^{\ast })\hookrightarrow L_{(0,q)}^2(\mathrm{\Omega })$ is continuous in $W^s(\mathrm{\Omega })$–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 $n\ge 3$). In particular, $j_1$ is always regular in Sobolev norms in ${\mathbb{C}}^2$, notwithstanding the fact that $N_1$ need not be.