Sobolev inequalities and regularity of the linearized complex Monge-Ampère and Hessian equations

Wang, Jiaxiang; Zhou, Bin

doi:10.1090/tran/9275

Sobolev inequalities and regularity of the linearized complex Monge-Ampère and Hessian equations
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by Jiaxiang Wang and Bin Zhou;

Trans. Amer. Math. Soc. 378 (2025), 447-475

DOI: https://doi.org/10.1090/tran/9275

Published electronically: October 23, 2024

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Abstract:

Let $u$ be a smooth, strictly $k$-plurisubharmonic function on a bounded domain $\Omega \in \mathbb C^n$ with $2\leq k\leq n$. The purpose of this paper is to study the regularity of solution to the linearized complex Monge-Ampère and Hessian equations when the complex $k$-Hessian $H_k[u]$ of $u$ is bounded from above and below. We first establish an estimate of Green’s functions associated to the linearized equations. Then we prove a class of new Sobolev inequalities. With these inequalities, we use Moser’s iteration to investigate the a priori estimates of Hessian equations and their linearized equations, as well as the Kähler scalar curvature equation. In particular, we obtain the Harnack inequality for the linearized complex Monge-Ampère and Hessian equations under an extra integrability condition on the coefficients.

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