  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.
