A priori estimates for complex Hessian equations

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 Kähler manifolds. We also show optimal $L^p$ integrability for $m$-subharmonic functions with compact singularities, thus partially confirming a conjecture of Błocki. 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.