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Proceedings of the American Mathematical Society

Published by the American Mathematical Society, the Proceedings of the American Mathematical Society (PROC) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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Continuity of translation operators
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by Krishna B. Athreya and Justin R. Peters PDF
Proc. Amer. Math. Soc. 139 (2011), 4027-4040 Request permission

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. $m$.
References
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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
  • © Copyright 2011 American Mathematical Society
  • 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
  • MathSciNet review: 2823048