Rate of equidistribution for the unstable manifolds of Anosov diffeomorphisms

Let $M$ be a compact Riemannian manifold. For a $C^3$ smooth topologicaly mixing Anosov diffeomorphism $F:M\rightarrow M$, we study the equidustribution properties of the unstable manifolds with respect to the Margulis measure of maximal entropy $\mathbf m$ Extending the results of Bufetov and Bufetov-Forni on geodesic/horocycle flows on compact Riemann surfaces of constant negative curvature to a non-linear setting, we prove that, under certain bounded distortion assumptions on the diffeomorphism, the leafwise averages on the unstable leaves of a $C^2$ smooth function $\psi:M\rightarrow\mathbb R$ with $\mathbf m(\psi)=0$ are controlled by a finitely additive measure on the unstable foliation, invariant under the holonomy along stable leaves.
Using the method Gou ̈ezel and Liverani, we contruct a Banach space of currents which admits an $F$-invariant finite dimensional subspace whose elements induce holonomy invariant finitely additive measures