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Sobolev Inequalities and the \(\overline{\partial }\)-Neumann Operator

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Abstract

We study a complex-valued version of the Sobolev inequalities and its relationship to compactness of the \(\overline{\partial }\)-Neumann operator. For this purpose we use an abstract characterization of compactness derived from a general description of precompact subsets in \(L^2\)-spaces. Finally we remark that the \(\overline{\partial }\)-Neumann operator can be continuously extended provided a subelliptic estimate holds.

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Acknowledgments

The author wishes to express his gratitude to the referee for helpful suggestions. Partially supported by the FWF-Grant P23664.

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  1. Fakultät für Mathematik, Universität Wien, Oskar-Morgenstern-Platz 1, 1090, Vienna, Austria

    Friedrich Haslinger

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Haslinger, F. Sobolev Inequalities and the \(\overline{\partial }\)-Neumann Operator. J Geom Anal 26, 287–293 (2016). https://doi.org/10.1007/s12220-014-9549-3

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