Which moments of a logarithmic derivative imply quasiinvariance (Q1271413)

Skorohod has shown that if a measure \(\mu\) on a Hilbert space has a logarithmic derivative \(\beta\) in a given direction \(h\) and \(\exp (a| \beta|)\) is integrable with respect to \(\mu\) for some positive \(a\), then \(\mu\) is quasi-invariant in the direction \(h\). In the present paper, the authors extend Shorohod's result to one-parameter families of measures. They also characterize the class of functions \(\psi: [0, \infty) \to [0, \infty)\) such that the integrability of \(\psi(| \beta |)\) implies quasi-invariance of \(\mu\). They show that if \(\psi\) is convex, then a necessary and sufficient condition is that the function \(\log \psi (x)/x^2\) is not integrable at \(\infty\).