  We are motivated by a conjecture of A. and S. Katok to study the smooth
cohomologies of a family of Weyl chamber flows. The conjecture is a natural
generalization of the Livshitz Theorem to Anosov actions by higher-rank abelian
groups; it involves a description of top-degree cohomology and a vanishing
statement for lower degrees. Our main result, proved in Part II, verifies the
conjecture in lower degrees for our systems, and steps in the "correct"
direction in top degree. In Part I we study our "base case": geodesic flows of
finite-volume hyperbolic manifolds. We describe obstructions (invariant
distributions) to solving the coboundary equation in unitary representations of
the group of orientation-preserving isometries of hyperbolic N-space, and we
study Sobolev regularity of solutions. (One byproduct is a smooth Livshitz
Theorem for geodesic flows of hyperbolic manifolds with cusps.) Part I provides
the tools needed in Part II for the main theorem.
