We introduce a notion of differential of a Sobolev map between metric spaces. The differential is given in the framework of tangent and cotangent modules of metric measure spaces, developed by the first author. We prove that our notion is consistent with Kirchheim's metric differential when the source is a Euclidean space, and with the abstract differential provided by the first author when the target is R.

We show that, given a metric space $$(\mathrm{Y},\textsf {d} )$$ ( Y , d ) of curvature bounded from above in the sense of Alexandrov, and a positive Radon measure $$\mu $$ μ on $$\mathrm{Y}$$ Y giving finite mass to bounded sets, the resulting metric measure space $$(\mathrm{Y},\textsf {d} ,\mu )$$ ( Y , d , μ ) is infinitesimally Hilbertian, i.e. the Sobolev space $$W^{1,2}(\mathrm{Y},\textsf {d} ,\mu )$$ W 1 , 2 ( Y , d , μ ) is a Hilbert space. The result is obtained by constructing an isometric embedding of the ‘abstract and analytical’ space of derivations into the ‘concrete and geometrical’ bundle whose fibre at $$x\in \mathrm{Y}$$ x ∈ Y is the tangent cone at x of $$\mathrm{Y}$$ Y . The conclusion then follows from the fact that for every $$x\in \mathrm{Y}$$ x ∈ Y such a cone is a $$\mathrm{CAT}(0)$$ CAT ( 0 ) space and, as such, has a Hilbert-like structure.

We consider functions with an asymptotic mean value property, known to characterize harmonicity in Riemannian manifolds and in doubling metric measure spaces. We show that the strongly amv-harmonic functions are Hölder continuous for any exponent below one. More generally, we define the class of functions with finite amv-norm and show that functions in this class belong to a fractional Hajłasz–Sobolev space and their blow-ups satisfy the mean-value property. Furthermore, in the weighted Euclidean setting we find an elliptic PDE satisfied by amv-harmonic functions.

No abstract