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Continuity of translation operators


Authors: Krishna B. Athreya and Justin R. Peters
Journal: Proc. Amer. Math. Soc. 139 (2011), 4027-4040
MSC (2010): Primary 26A42; Secondary 28A25, 22F10
DOI: https://doi.org/10.1090/S0002-9939-2011-10862-1
Published electronically: March 28, 2011
MathSciNet review: 2823048
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Abstract: For a Radon measure $\mu$ on $\mathbb {R},$ we show that $L^{\infty }(\mu )$ is invariant under the group of translation operators $T_t(f)(x) = {f(x-t)}\ (t \in \mathbb {R})$ if and only if $\mu$ is equivalent to the Lebesgue measure $m$. We also give necessary and sufficient conditions for $L^p(\mu ),\ 1 \leq p < \infty ,$ to be invariant under the group $\{ T_t\}$ in terms of the Radon-Nikodým derivative w.r.t. $m$.


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Additional Information

Krishna B. Athreya
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: kba@iastate.edu

Justin R. Peters
Affiliation: Department of Mathematics, Iowa State University, Ames, Iowa 50011
Email: peters@iastate.edu

Received by editor(s): March 18, 2010
Received by editor(s) in revised form: September 30, 2010
Published electronically: March 28, 2011
Additional Notes: The second author acknowledges partial support from the National Science Foundation, DMS-0750986
Communicated by: Thomas Schlumprecht
Article copyright: © Copyright 2011 American Mathematical Society