论文标题
非平滑设置中的部分衍生物
Partial derivatives in the nonsmooth setting
论文作者
论文摘要
我们研究了两个度量测量结构的乘积的部分衍生物，尤其是通过第一个命名作者提出的模块与微积分有关。 我们的主要结果是1）施瓦茨关于混合第二个衍生物的对称性的非平滑框架的扩展； 2）一组完整的结果，将属性$ f \在w^{2,2}（\ x \ x \ times \ y）中与$ f（\ cdot，y cdot，y）和$ f（\ cdot，y）和$ f（ $ f（x，\ cdot）\ in w^{2,2}（\ y）$ for A.E. \ $ y，x $在另一个方面分别为x $。这里$ \ x，\ y $是$ \ rcd $ spaces，以便二阶Sobolev空格得到很好的定义。 \ end {inatize}这些结果又基于对sobolev的规律性的研究和差异的基本概念的研究，对于希尔伯特模块中的值的地图：我们主要将此概念应用于地图$ x \ x \ mapsto \ d_ \ d_ \ d_ \ d_ \ d_ \ sy f（x，x，\ cdot）在适当的常规$下构建，sy d _ sy d \ s sy d \ s d \ s d \ s。 f $。
We study partial derivatives on the product of two metric measure structures, in particular in connection with calculus via modules as proposed by the first named author. Our main results are 1) The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivatives, 2) a quite complete set of results relating the property $f\in W^{2,2}(\X\times\Y)$ on one side with that of $f(\cdot,y)\in W^{2,2}(\X)$ and $f(x,\cdot)\in W^{2,2}(\Y)$ for a.e.\ $y,x$ respectively on the other. Here $\X,\Y$ are $\RCD$ spaces so that second order Sobolev spaces are well defined. \end{itemize} These results are in turn based upon the study of Sobolev regularity, and of the underlying notion of differential, for a map with values in a Hilbert module: we mainly apply this notion to the map $x\mapsto\d_\sy f(x,\cdot)$ in order to build, under the appropriate regularity requirements, its differential $\d_\sx\d_\sy f$.
