2018/11/08 by Debraj Chakrabarti, Chakrabarti, Debraj, Phillip S. Harrington +1
Mathematics · #Advanced Harmonic Analysis Research #Complex Variables (math.CV) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Holomorphic and Operator Theory
paper · pdf · doi:10.48550/arxiv.1811.03715
openalex publication_date 2018/11/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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 C1,1 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 C1,1 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.