= Solution
The <Livsic theorem> for a topologically transitive <Anosov flow> says that a Hölder function $h:N\to\mathbb R$ satisfies the periodic-orbit obstruction
$$
\int_0^T h(\phi_t x)\,dt=0\quad\text{whenever }\phi_Tx=x
$$
if and only if there is a Hölder function $u$ satisfying the flow coboundary identity
$$
\boxed{u(\phi_t x)-u(x)=\int_0^t h(\phi_sx)\,ds}.
$$
The 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.
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