  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 $p$.
