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Partial derivatives in the nonsmooth setting

2022; Elsevier BV; Volume: 283; Issue: 4 Linguagem: Inglês

10.1016/j.jfa.2022.109528

1096-0783

Nicola Gigli, Chiara Rigoni,

Nonlinear Partial Differential Equations

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: The extension to this non-smooth framework of Schwarz's theorem about symmetry of mixed second derivatives 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.