Lipschitz continuity of harmonic maps between ${\rm RCD}(K,N)$ spaces and ${\rm CAT}(\kappa)$ spaces

We are going to prove that energy minimizing harmonic maps from a domain $\Omega$ inside an ${\rm RCD}(K,N)$ whose image lies in a small ball inside a ${\rm CAT}(\kappa)$ space are locally Lipschitz continuous.

This completes the picture concerning regularity of harmonic maps between singular spaces and justifies a variant of the Bochner-Eells-Sampson inequality between singular spaces.