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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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  Jöran Bergh and Jörgen Löfström, Interpolation spaces. An introduction, Springer-Verlag, Berlin-New York, 1976. Grundlehren der Mathematischen Wissenschaften, No. 223. · Zbl 0344.46071  Max Bézard, Régularité \?^{\?} précisée des moyennes dans les équations de transport, Bull. Soc. Math. France 122 (1994), no. 1, 29 – 76 (French, with English and French summaries). · Zbl 0798.35025  Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026  Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018  R. A. DeVore, P. Petrushev, and X. M. Yu, Nonlinear wavelet approximation in the space \?(\?^{\?}), Progress in approximation theory (Tampa, FL, 1990) Springer Ser. Comput. Math., vol. 19, Springer, New York, 1992, pp. 261 – 283. · Zbl 0802.42027  Ronald A. DeVore and Robert C. Sharpley, Besov spaces on domains in \?^{\?}, Trans. Amer. Math. Soc. 335 (1993), no. 2, 843 – 864. · Zbl 0766.46015  R. J. DiPerna, P.-L. Lions, and Y. Meyer, \?^{\?} regularity of velocity averages, Ann. Inst. H. Poincaré Anal. Non Linéaire 8 (1991), no. 3-4, 271 – 287 (English, with French summary). · Zbl 0763.35014  François Golse, Pierre-Louis Lions, Benoît Perthame, and Rémi Sentis, Regularity of the moments of the solution of a transport equation, J. Funct. Anal. 76 (1988), no. 1, 110 – 125. · Zbl 0652.47031  Pierre-Louis Lions, Régularité optimale des moyennes en vitesses, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 8, 911 – 915 (French, with English and French summaries). · Zbl 0922.35135  Yves Meyer, Ondelettes et opérateurs. I, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Ondelettes. [Wavelets]. · Zbl 0694.41037  J. Peetre, A Theory of Interpolation Spaces, Notes, Universidade de Brasilia, 1963. · Zbl 0162.44502  G. Petrova, Transport Equations and Velocity Averages, Ph.D. Thesis, University of South Carolina, 1999.
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