Which moments of a logarithmic derivative imply quasiinvariance
Documenta mathematica, Tome 3 (1998), pp. 261-272
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 on a Hilbert space is quasiinvariant in a given direction if it has a logarithmic derivative β in this direction such that ea∣β∣ is μ-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 ψ:[0,∞)→[0,∞) for which the integrability of ψ(∣β∣) implies quasiinvariance of μ. If ψ is convex then a necessary and sufficient condition is that logψ(x)/x2 is not integrable at ∞.
@article{10_4171_dm_43,
author = {Heinrich von Weizs\"acker and Michael Scheutzow},
title = {Which moments of a logarithmic derivative imply quasiinvariance},
journal = {Documenta mathematica},
pages = {261--272},
year = {1998},
volume = {3},
doi = {10.4171/dm/43},
url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/43/}
}
Heinrich von Weizsäcker; Michael Scheutzow. Which moments of a logarithmic derivative imply quasiinvariance. Documenta mathematica, Tome 3 (1998), pp. 261-272. doi: 10.4171/dm/43
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