The Hölder continuous subsolution theorem for complex Hessian equations

Amel Benali,Ahmed Zeriahi

doi:10.5802/jep.133

Abstract

Let $\Omega \Subset \mathbb C^n$ be a bounded strongly $m$-pseudoconvex domain ($1\leq m\leq n$) and $\mu$ a positive Borel measure with finite mass on $\Omega$. Then we solve the H\"older continuous subsolution problem for the complex Hessian equation $(dd^c u)^m \wedge \beta^{n - m} = \mu$ on $\Omega$. Namely, we show that this equation admits a unique H\"older continuous solution on $\Omega$ with a given H\"older continuous boundary values if it admits a H\"older continuous subsolution on $\Omega$. The main step in solving the problem is to establish a new capacity estimate showing that the $m$-Hessian measure of a H\"older continuous $m$-subharmonic function on $\Omega$ with zero boundary values is dominated by the $m$-Hessian capacity with respect to $\Omega$ with an (explicit) exponent $\tau > 1$.

Highlights

Complex Hessian equations are important examples of fully non-linear PDE’s of second order on complex manifolds
Idea of the proof of Theorem B. — The proof will be in two steps. – The first step relies on a standard method which goes back to [EGZ09] in the case of the complex Monge-Ampère equation. This method consists in proving a semi-stability inequality estimating supΩ(v − u)+ in terms of (v − u)+ L1(Ω,μ), where u is the bounded m-subharmonic solution to the Dirichlet problem (1.1) and v is any bounded m-subharmonic function with the same boundary values as u, under the assumption that the measure μ is dominated by the m-Hessian capacity with an exponent τ > 1
– The second step uses an idea which goes back to [DDG+14] in the setting of compact Kähler manifolds. It has been used in the local setting in [Ngu18] and [KN20a]. It consists in estimating the L1(μ)-norm of v − u in terms of the L1(λ2n)-norm of (v − u) where u is the bounded solution to the Dirichlet problem and v is a bounded m-subharmonic function on Ω close to the regularization uδ of u

Summary

Introduction

Complex Hessian equations are important examples of fully non-linear PDE’s of second order on complex manifolds. – The first step relies on a standard method which goes back to [EGZ09] (see [GKZ08]) in the case of the complex Monge-Ampère equation This method consists in proving a semi-stability inequality estimating supΩ(v − u)+ in terms of (v − u)+ L1(Ω,μ), where u is the bounded m-subharmonic solution to the Dirichlet problem (1.1) and v is any bounded m-subharmonic function with the same boundary values as u, under the assumption that the measure μ is dominated by the m-Hessian capacity with an exponent τ > 1 (see Definition 2.18). It consists in estimating the L1(μ)-norm of v − u in terms of the L1(λ2n)-norm of (v − u) where u is the bounded solution to the Dirichlet problem and v is a bounded m-subharmonic function on Ω close to the regularization uδ of u This step requires that the measure μ is well dominated by the m-Hessian capacity, which is precisely the content of our Theorem A.

Preliminary results

An obstacle problem

Hessian measures of Hölder continuous potentials

Proofs of the main results