## The averaging lemma

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- by Ronald DeVore and Guergana Petrova
- J. Amer. Math. Soc.
**14**(2001), 279-296 - DOI: https://doi.org/10.1090/S0894-0347-00-00359-3
- Published electronically: November 30, 2000
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## 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 $p=1$.## References

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## Bibliographic 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