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 Cⁿ, where the inner domain has C^{1 , 1} boundary, we show that the L² 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 L² Sobolev space W¹ 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.