非光滑背景下的偏导数
Partial derivatives in the nonsmooth setting
摘要：
我们研究了两个度量测度结构的乘积的偏导数，特别是与第一作者提出的通过模块的微积分有关。我们的主要结果是：（1）推广了Schwarz关于混合二阶导数对称性的定理的非光滑框架;（2）对于a.e.给出了一组相当完整的结果，它们把一边的性质$f\in W^{2,2}(\X\times\Y)$与$f(\cdot,y)\in W^{2,2}(\X)$和$f(x,\cdot)\in W^{2,2}(\Y)$联系起来。$y,x$分别在另一侧。这里$\X,\Y$是$\RCD$空间，所以二阶Sobolev空间是很好定义的。\end{itemize}这些结果又是基于研究索伯列夫正则性，以及微分的基本概念，为一个地图的价值观在希尔伯特模块：我们主要应用这一概念的地图$x\mapsto\d_\sy f(x,\cdot)$，以建立在适当的正则性要求，其微分$\d_\sx\d_\sy f$。
英文摘要：
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 $\d_\sx\d_\sy f$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
