Reverse hypercontractivity over manifolds (Q1810321)

Consider a vector field \(X\) over a manifold \(M\) whose flow \(\exp tX\) exists for any time. Let \(\mu\) be a measure on \(M\) for which the induced measures \(\mu_t\equiv(\exp tX)_*\mu\) are absolutely continuous with respect to \(\mu\). The authors establish bounds for the \(L^p(\mu)\) norm of the Radon-Nikodym derivative \(d\mu_t/d\mu\) in terms of the divergence of the field \(X\). In the case of a complex manifold \(M\) they derive reverse hypercontractivity bounds and reverse logarithmic Sobolev inequalities in some holomorphic function spaces. Examples are proposed on \({\mathbb C}^m\) and on the Riemann surface for \(z^{1/n}\).