Velocity averaging -- a general framework

Details Velocity averaging -- a general framework Velocity averaging -- a general framework

2011-07-13

arxiv.org/abs/1107.2616v1

Mathematics

Analysis of PDEs

15 pages, 1 figure

Scientific paper

We prove that a sequence of averaged quantities $\int_{\R^m}h_n(x,p)\rho(p)dp$, $n\in \N$, is strongly precompact in $L^1_{loc}(\R^d)$, where $\rho\in C_0(\R^m)$, and $h_n\in L^p(\R^d\times \R^m)$, $p\geq 2$, are weak solutions to multiplier-operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators (which we provide as an example). If $p>2$ then the coefficients can be discontinuous with respect to the space variable $x\in \R^d$, otherwise, the coefficients are continuous functions. In order to obtain the result we prove a representation theorem for an extension of the $H$-measures.

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