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Transference for amenable actions

Authors: Waldemar Hebisch and M. Gabriella Kuhn
Journal: Proc. Amer. Math. Soc. 133 (2005), 1733-1740
MSC (2000): Primary 47A30; Secondary 37A15, 43A07
Published electronically: November 19, 2004
MathSciNet review: 2120272
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Abstract: Suppose $G$ acts amenably on a measure space $X$ with quasi-invariant $\sigma$-finite measure $m$. Let $\sigma$ be an isometric representation of $G$ on $L^p(X,dm)$ and $\mu$ a finite Radon measure on $G$. We show that the operator $\sigma_\mu f(x)=\int_G(\sigma(g)f)(x)d\mu(g)$ has $L^p(X,dm)$-operator norm not exceeding the $L^p(G)$-operator norm of the convolution operator defined by $\mu$. We shall also prove an analogous result for the maximal function $M$ associated to a countable family of Radon measures $\mu_n$.

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Additional Information

Waldemar Hebisch
Affiliation: Mathematical Institute, Wrocław University, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

M. Gabriella Kuhn
Affiliation: Dipartimento di Matematica, Università di Milano “Bicocca”, Via R. Cozzi 53, Edificio U5, 20125 Milano, Italia

Received by editor(s): July 19, 2003
Received by editor(s) in revised form: February 15, 2004
Published electronically: November 19, 2004
Additional Notes: The first author was supported by KBN: 5 P03A 050 20 and RTN: HPRN-CT-2001-00273, and partially by GNAMPA
Communicated by: Andreas Seeger
Article copyright: © Copyright 2004 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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