The averaging lemma

DeVore, Ronald¹ ; Petrova, Guergana²

¹ Department of Mathematics, University of South Carolina, Columbia, South Carolina 29208
² Department of Mathematics, University of Michigan, Ann Arbor, Michigan 48109

Journal of the American Mathematical Society, Tome 14 (2001) no. 2, pp. 279-296

Résumé

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 $p=1$.

Détail

DOI : 10.1090/S0894-0347-00-00359-3

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