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

DOI:
https://doi.org/10.1090/S0894-0347-00-00359-3

Published electronically:
November 30, 2000

MathSciNet review:
1815213

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Averaging lemmas deduce smoothness of velocity averages, such as

from properties of . A canonical example is that is in the Sobolev space whenever and are in . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet decompositions, and interpolation can be used to prove versions of the averaging lemma. For example, it is shown that implies that is in the Besov space , . 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 .

**1.**J. Bergh and J. Löfström,*Interpolation Spaces*, Springer-Verlag, 1976. MR**58:2349****2.**M. Bezard,*Regularite 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*, 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*, TAMS**335**(1993), 843-864. MR**93d:46051****7.**R. DiPerna, P-L. Lions, and Y. Meyer,*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.

Retrieve articles in *Journal of the American Mathematical Society*
with MSC (1991):
35L60,
35L65,
35B65,
46B70,
46B45,
42B25

Retrieve articles in all journals with MSC (1991): 35L60, 35L65, 35B65, 46B70, 46B45, 42B25

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:
https://doi.org/10.1090/S0894-0347-00-00359-3

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