  We introduce a refined Sobolev scale on a vector bundle over a closed
infinitely smooth manifold. This scale consists of inner product H\"ormander
spaces parametrized with a real number and a function varying slowly at
infinity in the sense of Karamata. We prove that these spaces are obtained by
the interpolation with a function parameter between inner product Sobolev
spaces. An arbitrary classical elliptic pseudodifferential operator acting
between vector bundles of the same rank is investigated on this scale. We prove
that this operator is bounded and Fredholm on pairs of appropriate H\"ormander
spaces. We also prove that the solutions to the corresponding elliptic equation
satisfy a certain a priori estimate on these spaces. The local regularity of
these solutions is investigated on the refined Sobolev scale. We find new
sufficient conditions for the solutions to have continuous derivatives of a
given order.
