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Journal of the American Mathematical Society
Journal of the American Mathematical Society
ISSN 1088-6834(online) ISSN 0894-0347(print)

 

The averaging lemma


Authors: Ronald DeVore and Guergana Petrova
Journal: J. Amer. Math. Soc. 14 (2001), 279-296
MSC (1991): Primary 35L60, 35L65, 35B65, 46B70; Secondary 46B45, 42B25
Published electronically: November 30, 2000
MathSciNet review: 1815213
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Abstract:

Averaging lemmas deduce smoothness of velocity averages, such as

\begin{displaymath}\bar f(x):=\int_\Omega f(x,v)\, dv ,\quad \Omega\subset \mathbb{R}^d, \end{displaymath}

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

DOI: http://dx.doi.org/10.1090/S0894-0347-00-00359-3
PII: S 0894-0347(00)00359-3
Keywords: Averaging lemma, regularity, transport equations, Besov spaces
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.
Article copyright: © Copyright 2000 American Mathematical Society