On Weighted \(L^{p}\)-Sobolev Estimates for Solutions of the \(\overline{\partial }\)-equation on Linearly Convex Domains of Finite Type and Application

We obtain some weighted \(L^{p}\) $L^p$ -Sobolev estimates with gain on p and the weight for solutions of the \(\overline{\partial }\) $\overline{\partial }$ -equation in linearly convex domains of finite type in \(\mathbb {C}^{n}\) ${\mathbb{C}}^n$ and apply them to obtain weighted \(L^{p}\) $L^p$ -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.