Partial derivatives in the nonsmooth setting

TL;DR

This paper extends calculus concepts like Schwarz's theorem to nonsmooth metric measure spaces, establishing relationships between second-order Sobolev regularity of functions on product spaces and their slices.

Contribution

It introduces a framework for partial derivatives in nonsmooth settings, including symmetry of mixed derivatives and Sobolev regularity relations on product spaces of RCD spaces.

Findings

Extension of Schwarz's theorem to nonsmooth metric measure spaces

Characterization of second-order Sobolev regularity on product spaces

Analysis of differential structures for functions valued in Hilbert modules

Abstract

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 $\d_\sx\d_\sy f$.