 Solution

The Livsic theorem for a topologically transitive Anosov flow says that a Hölder function $h:N\to \mathbb{R}$ satisfies the periodic-orbit obstructionif and only if there is a Hölder function $u$ satisfying the flow coboundary identityThe solution is differentiable along orbits and $Fu=h$ in that sense. For a smooth flow and smooth $h$, the smooth regularity version gives $u\in C^{\infty }(N)$, so $Fu=h$ holds smoothly. The solution is unique up to an additive constant: the difference of two solutions is invariant and must be constant on a dense orbit and then everywhere by continuity. Necessity of the periodic obstruction follows immediately by taking $t=T$ in the displayed identity.