Liouville and Calabi-Yau type theorems for complex Hessian equations

We prove a Liouville type theorem for entire maximal $m$-subharmonic functions in ${\mathbb{C}}^n$ with bounded gradient. This result, coupled with a standard blow-up argument, yields a (non-explicit) a priori gradient estimate for the complex Hessian equation on a compact K"ahler manifold. This terminates the program, initiated by Hou, Ma and Wu, of solving the non-degenerate Hessian equation on such manifolds in full generality. We also obtain, using our previous work, continuous weak solutions in the degenerate case for the right hand side in some $L^p,$ with sharp bound on $p$.