A remark on the Hölder regularity of solutions to the complex Hessian equation
Slawomir Kolodziej, Ngoc Cuong Nguyen
TL;DR
This work proves that the Dirichlet problem for the complex Hessian equation admits a Hölder continuous solution whenever a Hölder continuous $m$-subharmonic subsolution exists and the right-hand side measure $\mu$ is dominated by the Hessian measure $H_m(\varphi)$, without the finite total mass restriction. Building a capacity-dominated framework, it extends the subsolution to a neighborhood using a defining function and derives a key capacity estimate $\int_K H_m(\varphi) \le A_1 [cap_m(K)]^{1+\alpha_0}$ for all compact $K$, even for unbounded $\mu$ by splitting the domain and employing comparison principles. The Hölder regularity of the solution is then obtained by following Ng20's strategy, combining the capacity bounds with regularization, stability estimates, and a delicate L1–L1 control to obtain an explicit Hölder exponent $\alpha'$ depending on $\alpha$, $m$, and $n$. This extends the scope of Hölder regularity results beyond finite mass measures and without viscosity methods or rooftop envelopes, broadening the applicability of complex Hessian equation theory to more singular right-hand sides.
Abstract
We prove that the Dirichlet problem for the complex Hessian equation has the Hölder continuous solution provided it has a subsolution with this property. Compared to the previous result of Benali-Zeriahi and Charabati-Zeriahi we remove the assumption on the finite total mass of the measure on the right hand side.