A general method of weights in the d-bar-Neumann problem

Abstract

This thesis deals with Partial Differential Equations in Several Complex Variables and especially focuses on a general estimate for the $\bar\partial$-Neumann problem on a domain which is $q$-pseudoconvex or $q$-pseudoconcave at a boundary point $z_0$. Generalizing Property ($P$) by \cite{C84}, we define Property $(f\T-\M\T-P)^k$ at $z_0$. This property yields the estimate {(f\T-\M)^k} \qquad \no{f(\Lambda)\mathcal M u}^2\le c(\no{\bar\partial u}^2+\no{\bar\partial^*u}^2+\no{u}^2)+C_\M\no{u}^2_{-1} for any $u\in C^\infty_c(U\cap \bar{\Omega})^k\cap \T{Dom}(\dib^*)$ where $U$ is a neighborhood of $z_0$. We want to point out that under a suitable choice of $f$ and $\M$, $(f\T-\M)^k$ is the subelliptic, superlogarithmic, compactness and subelliptic multiplier estimate. The thesis also aims at exhibiting some relevant classes of domains which enjoy Property $(f\T-\M\T-P)^k$ and at discussing recent literature on the $\bar\partial$-Neumann problem in the framework of this property.