A priori estimates for complex Hessian equations

We prove some [math] a priori estimates as well as existence and stability theorems for the weak solutions of the complex Hessian equations in domains of [math] and on compact Kähler manifolds. We also show optimal [math] integrability for [math] -subharmonic functions with compact singularities, thus partially confirming a conjecture of Błocki. Finally we obtain a local regularity result for [math] 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.

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