Which Moments of a Logarithmic Derivative Imply Quasiinvariance?

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 \cite{Skorokhod74} 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 \cite{Smolyanov-Weizsaecker93} to extend this result to general one-parameter families of measures and moreover we give a complete characterization of all functions $\psi:[0,\infty) \rightarrow [0,\infty)$ 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$.

1991 Mathematics Subject Classification: 26 A 12, 28 C 20, 60 G 30

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