  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.
