Reverse hypercontractivity over manifolds

Suppose that X is a vector field on a manifold M whose flow, exp tX, exist, for all time. If mu is a measure on M for which the induced measures mu(t) = (exp t X)(*)mu are absolutely continuous with respect to mu. it is of interest to establish bounds on the L-p(mu) norm of the Radon Nikodym derivative dmu(t)/dmu. We establish such bounds in terms of the divergence of the vector field X. We then specialize M to he a complex manifold and derive reverse hypercontractivity hounds and reverse logarithmic Sobolev inequalities in some holomorphic function spaces. We give examples on C-m and on the Riemann surface for z(1/n).