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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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Characterizations of bounded mean oscillation
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by Stephen Jay Berman PDF
Proc. Amer. Math. Soc. 51 (1975), 117-122 Request permission

Abstract:

Recall that an integrable function $f$ on a cube ${Q_0}$ in ${{\mathbf {R}}^n}$ is said to be of bounded mean oscillation if there is a constant $K$ such that for every parallel subcube $Q$ of ${Q_0}$ there exists a constant ${a_Q}$ such that $\int _Q {|f - {a_Q}| \leq K|Q|}$, where $|Q|$ denotes the volume of $Q$. We prove here that if there is an integer $d$ and a constant $K$ such that for every parallel subcube $Q$ of ${Q_0}$ there exists a polynomial ${p_Q}$ of degree $\leq d$ such that $\int _Q {|f - {p_Q}| \leq K|Q|}$, then $f$ is of bounded mean oscillation.
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Additional Information
  • © Copyright 1975 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 51 (1975), 117-122
  • MSC: Primary 42A92; Secondary 46E30
  • DOI: https://doi.org/10.1090/S0002-9939-1975-0374805-5
  • MathSciNet review: 0374805