Reverse hypercontractivity over manifolds

Abstract

Suppose that X is a vector field on a manifold M whose flow, exp tX, exists for all time. If μ is a measure on M for which the induced measures μ_t≡(exptX)_*μ are absolutely continuous with respect to μ, it is of interest to establish bounds on the L^p (μ) norm of the Radon-Nikodym derivative dμ_t/dμ. We establish such bounds in terms of the divergence of the vector field X. We then specilize M to be a complex manifold and derive reverse hypercontractivity bounds and reverse logarithmic Sololev inequalities in some holomorphic function spaces. We give examples on C^m and on the Riemann surface for z^1/n.