  For any $C^r$ contact Anosov flow with $r\ge 3$, we construct a scale of
Hilbert spaces, which are embedded in the space of distributions on the phase
space and contain all $C^r$ functions, such that the transfer operators for the
flow extend to them boundedly and that the extensions are quasi-compact.
Further we give explicit bounds on the essential spectral radii of the
extensions in terms of the differentiability r and the hyperbolicity exponents
of the flow.
