速度平均-一个一般框架
Velocity averaging -- a general framework
摘要：
证明了平均量序列$\int_{\R^m}u_n(\mx,\msnop)$ $ρ(\msnop)d\msnop$，在$\Ldl\Rd$中是强预紧的，其中$ρ\in \Ldc{\R^m}$，和$u_n\in \Ld{\R^m; \pL s\Rd}$，$s\geq 2$是变系数微分算子方程的弱解.特别地，这包括双曲型、抛物型或超抛物型的微分算子，但也包括分数微分算子。如果$s>2$，则系数可以相对于空间变量$\mx\in \R^d$不连续，否则，系数是连续函数。为了得到这个结果，我们证明了$H$ -测度的一个扩张的表示定理.
英文摘要：
We prove that the sequence of averaged quantities $\int_{\R^m}u_n(\mx,\msnop)$ $ρ(\msnop)d\msnop$, is strongly precompact in $\Ldl\Rd$, where $ρ\in \Ldc{\R^m}$, and $u_n\in \Ld{\R^m; \pL s\Rd}$, $s\geq 2$, are weak solutions to differential operator equations with variable coefficients. In particular, this includes differential operators of hyperbolic, parabolic or ultraparabolic type, but also fractional differential operators. If $s>2$ then the coefficients can be discontinuous with respect to the space variable $\mx\in \R^d$, 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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