Which moments of a logarithmic derivative imply quasiinvariance

Abstract

In many special contexts quasiinvariance of a measure under a one-parameter group of transformations has been established. A remarkable classical general result of A.V. Skorokhod citeSkorokhod74 states that a measure $\mu$ on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative $\beta$ in this direction such that $e^{a|\beta |}$ is $\mu$-integrable for some $a>0$. In this note we use the techniques of citeSmolyanov-Weizsaecker93 to extend this result to general one-parameter families of measures and moreover we give a complete characterization of all functions $\psi :[0,\infty )\to [0,\infty )$ for which the integrability of $\psi (|\beta |)$ implies quasiinvariance of $\mu$. If $\psi$ is convex then a necessary and sufficient condition is that $\log \psi (x)/x^2$ is not integrable at $\infty$.