Transference for amenable actions

Hebisch, Waldemar; Kuhn, M.

doi:10.1090/S0002-9939-04-07741-X

Transference for amenable actions
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by Waldemar Hebisch and M. Gabriella Kuhn PDF

Proc. Amer. Math. Soc. 133 (2005), 1733-1740 Request permission

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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