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Journal of the American Mathematical Society

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

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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The averaging lemma
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by Ronald DeVore and Guergana Petrova PDF
J. Amer. Math. Soc. 14 (2001), 279-296 Request permission

Abstract:

Averaging lemmas deduce smoothness of velocity averages, such as \[ \bar f(x):=\int _\Omega f(x,v) dv ,\quad \Omega \subset \mathbb {R}^d, \] from properties of $f$. A canonical example is that $\bar f$ is in the Sobolev space $W^{1/2}(L_2(\mathbb {R}^d))$ whenever $f$ and $g(x,v):=v\cdot \nabla _xf(x,v)$ are in $L_2(\mathbb {R}^d\times \Omega )$. The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove $L_p$ versions of the averaging lemma. For example, it is shown that $f,g\in L_p(\mathbb {R}^d\times \Omega )$ implies that $\bar f$ is in the Besov space $B_p^s(L_p(\mathbb {R}^d))$, $s:=\min (1/p,1/p^\prime )$. Examples are constructed using wavelet decompositions to show that these averaging lemmas are sharp. A deeper analysis of the averaging lemma is made near the endpoint $p=1$.
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Additional Information
  • Ronald DeVore
  • Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
  • Email: devore@math.sc.edu
  • Guergana Petrova
  • Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109
  • Email: petrova@math.lsa.umich.edu
  • Received by editor(s): November 18, 1999
  • Received by editor(s) in revised form: July 7, 2000
  • Published electronically: November 30, 2000
  • Additional Notes: Both authors were supported in part by the Office of Naval Research Contract N0014-91-J1343.
    The second author was also supported by the Rackham Grant and Fellowship Program.
  • © Copyright 2000 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 14 (2001), 279-296
  • MSC (1991): Primary 35L60, 35L65, 35B65, 46B70; Secondary 46B45, 42B25
  • DOI: https://doi.org/10.1090/S0894-0347-00-00359-3
  • MathSciNet review: 1815213