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arXiv · 1811.03715

A Modified Morrey-Kohn-H\"ormander Identity and Applications

Debraj Chakrabarti·Phillip S. Harrington
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Abstract

We prove a modified form of the classical Morrey-Kohn-H\"ormander 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 $1\leq q\leq n-2$ 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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Debraj Chakrabarti, Phillip S. Harrington. 2018-11-08. A Modified Morrey-Kohn-H\"ormander Identity and Applications. https://arxiv.org/abs/1811.03715

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