A Modified Morrey-Kohn-Hörmander Identity and Applications to the ∂¯-Problem

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 C n , where the inner domain has C 1 , 1 boundary, we show that the L 2 Dolbeault cohomology group in bidegree ( p , q ) vanishes if 1 ≤ q ≤ 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 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 .