The averaging lemma

Averaging lemmas arise in the study of regularity of solutions to nonlinear transport equations. The present paper shows how techniques from Harmonic Analysis, such as wavelet decompositions, maximal functions, and interpolation, can be used to prove averaging lemmas and to establish their sharpness. Let f(x, v) be a real-valued function defined on R × Ω, where Ω is a bounded domain in R. In applications Ω is a set of velocity vectors. Associated to f , we have the velocity average