  Liv\v{s}ic theorem asserts that, for Anosov diffeomorphisms/flows, a
Lipschitz observable is a coboundary if all its Birkhoff sums on every periodic
orbits are equal to zero. The transfer function is then Lipschitz. We prove a
positive Liv\v{s}ic theorem which asserts that a Lipschitz observable is
bounded from below by a coboundary if and only if all its Birkhoff sums on
periodic orbits are non negative. The new result is that the coboundary can be
chosen Lipschitz. The map is only assumed to be $C^1$ and hyperbolic, but not
necessarily bijective nor transitive. We actually prove our main result in the
setting of locally maximal hyperbolic sets for not general $C^1$ map. The
construction of the coboundary uses a new notion of the Lax-Oleinik operator
that is a standard tool in the discrete Aubry-Mather theory.
