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The averaging lemma. (English) Zbl 1001.35079
Summary: Averaging lemmas deduce smoothness of velocity averages, such as $\overline f(x):=\int_\Omega f(x,v) dv ,\quad \Omega\subset \mathbb{R}^d,$ from properties of $$f$$. A canonical example is that $$\overline f$$ 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 $$\overline f$$ is in the Besov space $$B_p^s(L_p(\mathbb{R}^d))$$, $$s:=\min(1/p,1/p')$$. 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 $$p=1$$.

##### MSC:
 35L60 First-order nonlinear hyperbolic equations 42B25 Maximal functions, Littlewood-Paley theory 35B65 Smoothness and regularity of solutions to PDEs 46B70 Interpolation between normed linear spaces 46B45 Banach sequence spaces 35L65 Hyperbolic conservation laws 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
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