Inserted: 7 dec 2020
Last Updated: 7 dec 2020
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: 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 under- lying notion of differential, for a map with values in a Hilbert module.