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.
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We thank Peter Ebenfelt and László Lempert for their valuable comments.
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Debraj Chakrabarti was partially supported by NSF Grant DMS-1600371.
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Chakrabarti, D., Harrington, P.S. A Modified Morrey-Kohn-Hörmander Identity and Applications to the \(\overline{\partial }\)-Problem. J Geom Anal 31, 9639–9676 (2021). https://doi.org/10.1007/s12220-021-00623-2
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DOI: https://doi.org/10.1007/s12220-021-00623-2