A new method to introduce a priori estimates for the ∂̄ Neumann problem

Abstract

0. A priori estimates for the ∂̄ Neumann problem have been induced by the Morrey trick up to now ( cf. [S] ). From this technique we can obtain an integral formula which contains a boundary integral whose integrand is the quadratic form induced from the Levi form ℒ (r) of a defining function r. To analyze the global regularity of Neumann operator, this boundary integral plays an important role. Here we introduce a quite different method to induce this integral formula. In fact we induce this formula from a kind of variation formula for the boundary integral of differential forms satisfying the first Neumann condition. This formula tells us the boundary behavior of the boundary integral which appears in the a priori estimate. As an application we can show a global regularity theorem for the Neumann operator on a pseudoconvex domain.