Weak solutions of complex Hessian equations on compact Hermitian manifolds

TL;DR

This paper establishes the existence of weak solutions to complex m-Hessian equations on compact Hermitian manifolds with nonnegative L^p data, extending previous results for smooth positive cases and providing stability insights.

Contribution

It proves the existence of weak solutions for complex m-Hessian equations with L^p data on Hermitian manifolds, generalizing prior smooth case solutions.

Findings

Existence of weak solutions for nonnegative L^p data

Stability results for these solutions

Extension of smooth case solutions to weaker data

Abstract

We prove the existence of weak solutions of complex $m-$Hessian equations on compact Hermitian manifolds for the nonnegative right hand side belonging to $L^p, p>n/m$ ($n$ is the dimension of the manifold). For smooth, positive data the equation has been recently solved by Szekelyhidi and Zhang. We also give a stability result for such solutions.