A priori estimates for the complex Hessian equations

TL;DR

This paper establishes key a priori estimates, existence, and stability results for solutions to complex Hessian equations, advancing understanding of their regularity and integrability properties on complex domains and Kähler manifolds.

Contribution

It provides new $L^{ abla}$ a priori estimates, proves existence and stability of weak solutions, and confirms a conjecture on optimal $L^p$ integrability for m-subharmonic functions.

Findings

Established $L^{ abla}$ a priori estimates for solutions.

Proved existence and stability theorems for weak solutions.

Confirmed optimal $L^p$ integrability for m-subharmonic functions with singularities.

Abstract

We prove some $L^{\infty}$ a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of $C^n$ and on compact K\"ahler manifolds. We also show optimal $L^p$ integrability for m-subharmonic functions with compact singularities, thus partially confirming a conjecture of Blocki. Finally we obtain a local regularity result for $W^{2,p}$ solutions of the real and complex Hessian equations under suitable regularity assumptions on the right hand side. In the real case the method of this proof improves a result of Urbas.