# The averaging lemma Academic Article

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### 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):=vcdot
abla _xf(x,v)
are in L 2 ( R d × Ω ) L_2(mathbb {R}^d imes Omega ) . The present paper shows how techniques from Harmonic

### author list (cited authors)

• DeVore, R., & Petrova, G.

### publication date

• January 1, 2000 11:11 AM