Comments on the regularity of harmonic maps between singular spaces

In this work we are going to establish H\"older continuity of harmonic maps from an open set Ω in an RCD(K,N) space valued into a CAT(κ) space, with the constraint that the image of Ω via the map is contained in a sufficiently small ball in the target. Building on top of this regularity and assuming a local Lipschitz regularity of the map, we establish a weak version of the Bochner-Eells-Sampson inequality in such a non-smooth setting. Finally we study the boundary regularity of such maps.