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Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations

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Abstract

In this paper, we study the relations between trace inequalities (Sobolev and Moser-Trudinger types), isocapacitary inequalities, and the regularity of the complex Hessian and Monge-Ampère equations with respect to a general nonnegative Borel measure. We obtain a quantitative characterization for these relations through the properties of the capacity-minimizing functions.

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Acknowledgements

The first author was supported by China Postdoctoral Science Foundation (Grant No. BX2021015). The second author was supported by National Key R&D Program of China (Grant No. SQ2020YFA0712800) and National Natural Science Foundation of China (Grant No. 11822101).

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Authors and Affiliations

  1. Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Jiaxiang Wang

  2. School of Mathematical Sciences, Peking University, Beijing, 100871, China

    Bin Zhou

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  1. Jiaxiang Wang
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  2. Bin Zhou
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Correspondence to Bin Zhou.

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Wang, J., Zhou, B. Trace inequalities, isocapacitary inequalities, and regularity of the complex Hessian equations. Sci. China Math. 67, 557–576 (2024). https://doi.org/10.1007/s11425-022-2100-1

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