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The Sobolev inequality for complex Hessian equations

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Abstract

In this paper, we study the complex Hessian equations by an gradient flow method. We prove a Sobolev inequality for \(k\)-plurisubharmonic functions analogous to that for real Hessian equations (Wang in Indiana Univ Math J 43:25–54, 1994; Lecture Notes in Mathematics, vol. 1977, 2009).

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Acknowledgments

The author is very grateful to X.-J. Wang for many valuable discussions. He would also like to thank the referee for pointing out several errors in an early version of this paper. This work was done when I was visiting the Simons Center for Geometry and Physics, Stony Brook University. I acknowledge the SCGP for the hospitality. I would also like to thank G. Tian and X.H. Zhu for their constant support.

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Authors and Affiliations

  1. School of Mathematical Sciences and Beijing International Center for Mathematical Research, Peking University, Beijing, 100871, China

    Bin Zhou

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  1. Bin Zhou
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Correspondence to Bin Zhou.

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B. Zhou was supported in part by National Science Foundation of China No. 11101004 and China Postdoctoral Science Foundation.

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Zhou, B. The Sobolev inequality for complex Hessian equations. Math. Z. 274, 531–549 (2013). https://doi.org/10.1007/s00209-012-1084-y

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