A sufficient condition for global regularity of the [formula omitted]-Neumann operator

Abstract

A theory of global regularity of the ∂ ¯ -Neumann operator is developed which unifies the two principal approaches to date, namely the one via compactness due to Kohn–Nirenberg [J.J. Kohn, L. Nirenberg, Non-coercive boundary value problems, Comm. Pure Appl. Math. 18 (1965) 443–492] and Catlin [David Catlin, Global regularity of the ∂ ¯ -Neumann problem, in: Y.-T. Siu (Ed.), Complex Analysis of Several Variables, in: Proc. Sympos. Pure Math., vol. 41, Amer. Math. Soc., Providence, RI, 1984, pp. 39–49] and the one via plurisubharmonic defining functions and/or vector fields that commute approximately with ∂ ¯ due to Boas and the author [Harold P. Boas, Emil J. Straube, Sobolev estimates for the ∂ ¯ -Neumann operator on domains in C n admitting a defining function that is plurisubharmonic on the boundary, Math. Z. 206 (1) (1991) 81–88; Harold P. Boas, Emil J. Straube, De Rham cohomology of manifolds containing the points of infinite type, and Sobolev estimates for the ∂ ¯ -Neumann problem, J. Geom. Anal. 3 (3) (1993) 225–235].