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 in an RCD (K, N) space valued into a CAT () space, with the constraint that the image of 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.