Which moments of a logarithmic derivative imply quasiinvariance

  • Michael Scheutzow

    Fachbereich Mathematik Fachbereich Mathematik der Technischen Universitat der Universitat Stra e des 17. Juni 135 D 67653 Kaiserslautern D 10623 Berlin Germany Germany
  • Heinrich von Weizsäcker

    Fachbereich Mathematik Fachbereich Mathematik der Technischen Universitat der Universitat Stra e des 17. Juni 135 D 67653 Kaiserslautern D 10623 Berlin Germany Germany
Which moments of a logarithmic derivative imply quasiinvariance cover
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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 μ\mu on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative β\beta in this direction such that eaβe^{a|\beta|} is μ\mu-integrable for some a>0a > 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 ψ:[0,)[0,)\psi:[0,\infty) \rightarrow [0,\infty) for which the integrability of ψ(β)\psi(|\beta|) implies quasiinvariance of μ\mu. If ψ\psi is convex then a necessary and sufficient condition is that logψ(x)/x2\log \psi(x)/{x^2} is not integrable at \infty.

Cite this article

Michael Scheutzow, Heinrich von Weizsäcker, Which moments of a logarithmic derivative imply quasiinvariance. Doc. Math. 3 (1998), pp. 261–272

DOI 10.4171/DM/43