On Weighted $L^p$-Sobolev Estimates for Solutions of the $\bar{\mathrm{\partial }}$-equation on Lineally Convex Domains of Finite Type and Application

We obtain some weighted $L^p$-Sobolev estimates with gain on $p$ and the weight for solutions of the $\bar{\mathrm{\partial }}$-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.