Suppose G acts amenably on a measure space X with quasiinvariant σ-finite measure m. Let σ be an isometric representation of G on L<sup>p</sup>(X, dm) and μ a finite Radon measure on G. We show that the operator σμf(x) = f<sub>G</sub>(σ(g)f)(x)dμ(g) has L <sub>p</sub>(X, dm)-operator norm not exceeding the L<sup>p</sup>(G)-operator norm of the convolution operator defined by μ. We shall also prove an analogous result for the maximal function M associated to a countable family of Radon measures μ<sub>n</sub>. © 2004 American Mathematical Society.
Hebisch, W., Kuhn, M. (2005). Transference for amenable actions. PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, 133(6), 1733-1740 [10.1090/S0002-9939-04-07741-X].
Transference for amenable actions
KUHN, MARIA GABRIELLA
2005
Abstract
Suppose G acts amenably on a measure space X with quasiinvariant σ-finite measure m. Let σ be an isometric representation of G on Lp(X, dm) and μ a finite Radon measure on G. We show that the operator σμf(x) = fG(σ(g)f)(x)dμ(g) has L p(X, dm)-operator norm not exceeding the Lp(G)-operator norm of the convolution operator defined by μ. We shall also prove an analogous result for the maximal function M associated to a countable family of Radon measures μn. © 2004 American Mathematical Society.
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