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$μ$_{$t$}≡(exp$tX$)_{*}$μ$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/$n$}.

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Let ϕ be a faithful normal semi-finite weight on a von Neumann algebra$M$. For each normal semi-finite weight ϕ on$M$, invariant under the modular automorphism group Σ of ϕ, there is a unique self-adjoint positive operator$h$, affiliated with the sub-algebra of fixed-points for Σ, such that ϕ=ϕ($h$·). Conversely, each such$h$determines a Σ-invariant normal semi-finite weight. An easy application of this non-commutative Radon-Nikodym theorem yields the result that$M$is semi-finite if and only if Σ consists of inner automorphisms.