The averaging lemma

Abstract

Averaging lemmas deduce smoothness of velocity averages, such as \[ f ¯ ( x ) := ∫ Ω f ( x , v ) d v , Ω ⊂ R d , \bar f(x):=\int _\Omega f(x,v)\, dv ,\quad \Omega \subset \mathbb {R}^d, \] from properties of f f . A canonical example is that f ¯ \bar f is in the Sobolev space W 1 / 2 ( L 2 ( R d ) ) W^{1/2}(L_2(\mathbb {R}^d)) whenever f f and g ( x , v ) := v ⋅ ∇ x f ( x , v ) g(x,v):=v\cdot abla _xf(x,v) are in L 2 ( R d × Ω ) 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 L_p versions of the averaging lemma. For example, it is shown that f , g ∈ L p ( R d × Ω ) f,g\in L_p(\mathbb {R}^d\times \Omega ) implies that f ¯ \bar f is in the Besov space B p s ( L p ( R d ) ) B_p^s(L_p(\mathbb {R}^d)) , s := min ( 1 / p , 1 / p ′ ) 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 p=1 .