Partial derivatives in the nonsmooth setting

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 in [13]. Our main results are: i) The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivatives ii) A quite complete set of results relating the property f∈W2,2(X×Y) on one side with that of f(⋅,y)∈W2,2(X) and f(x,⋅)∈W2,2(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. 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↦dyf(x,⋅) in order to build, under the appropriate regularity requirements, its differential dxdyf.