The averaging lemma Academic Article

Averaging lemmas deduce smoothness of velocity averages, such as [ f ( x ) := f ( x , v ) d v , 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):=vcdot
abla _xf(x,v) are in L 2 ( R d ) L_2(mathbb {R}^d imes Omega ) . The present paper shows how techniques from Harmonic Analysis such as maximal functions, wavelet deco