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$$L^ p$$ regularity of velocity averages. (English) Zbl 0763.35014
The paper is concerned with the regularity of velocity averages for solutions of transport equations: $v.\nabla_ xf=g\quad\text{for } x\in\mathbb{R}^ N,\quad v\in\mathbb{R}^ N,\quad\text{or} \tag{1}$
$\partial f/\partial t+v.\nabla_ xf=g\quad\text{for } x\in\mathbb{R}^ N,\quad v\in\mathbb{R}^ N,\quad t\in\mathbb{R}. \tag{2}$ In the time-independent case (equation (1)), the authors prove that, if $$f\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N)$$ and $$g\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N)$$, $$1<p\leq 2$$, then for every $$\psi\in D(\mathbb{R}^ N)$$, the velocity average $$\bar f(x)=\int_{\mathbb{R}^ N}f(x,v)\psi(v)dv$$ belongs to the Besov space $$B_ 2^{s,p}(\mathbb{R}^ N)$$ where $$s=1/p'$$.
In the time dependent case (equation (2)), they prove similar results when $$g$$ admits the following decomposition: $$g=(I-\Delta_ x)^{\tau/2}(I-\Delta_ v)^{m/2}G$$, $$\;\tau\in[0,1)$$, $$\;m\geq 0$$, $$\;G\in L^ p(\mathbb{R}^ N\times\mathbb{R}^ N\times\mathbb{R})$$.
Some applications to Vlasov-Maxwell systems and to other models are also given. These results extend many previous results [see for example F. Golse, P. L. Lions, B. Perthame, R. Sentis, J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)].

##### MSC:
 35B65 Smoothness and regularity of solutions to PDEs 35F05 Linear first-order PDEs 35Q40 PDEs in connection with quantum mechanics 42B25 Maximal functions, Littlewood-Paley theory 42B30 $$H^p$$-spaces 46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems 42B15 Multipliers for harmonic analysis in several variables 82B40 Kinetic theory of gases in equilibrium statistical mechanics 42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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