A General Estimate for the \(\bar \partial \)-Neumann Problem

Abstract

This paper especially focuses on a general estimate, called \((f-\mathcal M)^{k}\), for the \(\bar \partial \)-Neumann problem

\({(f-\mathcal M)^{k}} \qquad \| f({\varLambda })\mathcal M u\|^{2}\le c(\|\bar \partial u\|^{2}+\|\bar \partial ^{*}u\|^{2}+\|u\|^{2})+C_{\mathcal M}\|u\|^{2}_{-1}\)

for any \(u\in C^{\infty }_{c}(U\cap \bar {\Omega })^{k}\cap \text {Dom}(\bar {\partial }^{*})\), where f(Λ) is the tangential pseudodifferential operator with symbol f((1 + |ξ|²)^1/2), \(\mathcal M\) is a multiplier, and U is a neighborhood of a given boundary point z₀. Here the domain Ω is q-pseudoconvex or q-pseudoconcave at z₀. We want to point out that under a suitable choice of f and \(\mathcal M\), \((f{-}\mathcal M)^{k}\) is the subelliptic, superlogarithmic, compactness and so on. Generalizing the Property (P) by Catlin (1984), we define Property \((f-\mathcal M-P)^{k}\). The result we obtain in here is: Property \((f-\mathcal M-P)^{k}\) yields the \((f-\mathcal M)^{k}\) estimate. The paper also aims at exhibiting some relevant classes of domains which enjoy Property \((f-\mathcal M-P)^{k}\).