  We obtain some weighted $L^{p}$-Sobolev estimates with gain on $p$ and the
weight for solutions of the $\overline{\partial}$-equation in lineally convex
domains of finite type in $\mathbb{C}^{n}$ and apply them to obtain weighted
$L^{p}$-Sobolev estimates for weighted Bergman projections of convex domains of
finite type for quite general weights equivalent to a power of the distance to
the boundary.
