A Modified Morrey-Kohn-Hörmander Identity and Applications

Abstract

We prove a modified form of the classical Morrey-Kohn-Hörmander identity, adapted to pseudoconcave boundaries. Applying this result to an annulus between two bounded pseudoconvex domains in ${\mathbb{C}}^n$, where the inner domain has ${\mathcal{C}}^{1,1}$ boundary, we show that the $L^2$ Dolbeault cohomology group in bidegree $(p,q)$ vanishes if $1\le q\le n-2$ and is Hausdorff and infinite-dimensional if $q=n-1$, so that the Cauchy-Riemann operator has closed range in each bidegree. As a dual result, we prove that the Cauchy-Riemann operator is solvable in the $L^2$ Sobolev space $W^1$ on any pseudoconvex domain with ${\mathcal{C}}^{1,1}$ boundary. We also generalize our results to annuli between domains which are weakly $q$-convex in the sense of Ho for appropriate values of $q$.

Version 2: some minor typos have been fixed