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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 | References | Similar Articles | Additional Information


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 $p=1$.

References [Enhancements On Off] (What's this?)

  • 1. J. Bergh and J. Löfström, Interpolation Spaces, Springer-Verlag, 1976. MR 58:2349
  • 2. M. Bezard, Regularite $L^p$ precisee des moyennes dans les equations de transport, Bull. Soc. Math. France 22 (1994), 29-76. MR 95g:82083
  • 3. I. Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure and Appl. Math. 41 (1988), 909-996. MR 90m:42039
  • 4. I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. MR 93e:42045
  • 5. R. DeVore, P. Petrushev, and X. Yu, Wavelet approximation in the space $C(\mathbb{R}^d)$, Progress in Approximation Theory, Springer Verlag, New York, 1992, 261-283. MR 94h:41070
  • 6. R. DeVore and R. Sharpley, Besov spaces on domains in $\mathbb{R}^d$, TAMS 335 (1993), 843-864. MR 93d:46051
  • 7. R. DiPerna, P-L. Lions, and Y. Meyer, $L_p$ regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), 271-288. MR 92g:35036
  • 8. F. Golse, P-L. Lions, B. Perthame, and R. Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), 110-125. MR 89a:35179
  • 9. P-L. Lions, Regularite optimale des moyennes en vitesses, C. R. Acad. Sci. Paris 320 (1995), 911-915. MR 96c:35184
  • 10. Y. Meyer, Ondelettes et Opérateurs, Hermann, Paris, 1990. MR 93i:42002
  • 11. J. Peetre, A Theory of Interpolation Spaces, Notes, Universidade de Brasilia, 1963.
  • 12. G. Petrova, Transport Equations and Velocity Averages, Ph.D. Thesis, University of South Carolina, 1999.

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Additional Information

Ronald DeVore
Affiliation: Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208

Guergana Petrova
Affiliation: Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

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

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