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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

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
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

DOI: http://dx.doi.org/10.1090/S0002-9939-2011-10862-1
PII: S 0002-9939(2011)10862-1
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