A Modified Morrey-Kohn-Hörmander Identity and Applications to the $$\overline{\partial }$$-Problem

Abstract

We prove a modified form of the classical Morrey-Kohn-Hormander 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.