On the regularity of the complex Hessian equation

TL;DR

This paper investigates the regularity of solutions to the complex Hessian equation with specific density conditions, contributing to understanding solution smoothness under certain integrability constraints.

Contribution

It provides new regularity results for solutions to the complex Hessian equation when the measure density is in L^q for q ≤ n/m.

Findings

Established regularity criteria for solutions based on measure density in L^q.

Extended known results to cases with less regular density functions.

Identified conditions under which solutions exhibit improved regularity.

Abstract

This note aims to investigate the regularity of a solution to the Dirichlet problem for the complex Hessian equation, which has a density of the $m$-Hessian measure that belongs to $L^q$, for $q\leq\frac nm$.