  (Revised version, January 2006. S. Gouezel pointed out that, when 1<r<2, the
proof in the previous version was incomplete. In fixing this gap, we simplified
the argument in Section 6. In addition, there is a new appendix, with an
alternative description of the norms. We also corrected a few minor flaws.)
  We study spectral properties of transfer operators for diffeomorphisms T on a
Riemannian manifold: Suppose that there is an isolated hyperbolic subset for T,
with a compact isolating neighborhood V. We first introduce Banach spaces of
distributions supported on V, which are anisotropic versions of the usual space
of C^p functions C^p(V) and of the generalized Sobolev spaces W^{p,t}(V),
respectively. Then we show that the transfer operators associated to T and a
smooth weight extend boundedly to these spaces, and we give bounds on the
essential spectral radii of such extensions in terms of hyperbolicity
exponents. These bounds shed some light on those obtained by Kitaev for the
radius of convergence of dynamical determinants.
