The averaging lemma

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 $p=1$.