abstract: We establish Lipschitz regularity of harmonic maps from RCD (K, N) metric measure spaces with lower Ricci curvature bounds and dimension upper bounds in synthetic sense with values into CAT (0) metric spaces with non-positive sectional curvature. Under the same assumptions, we obtain a Bochner-Eells-Sampson inequality with a Hessian type-term. This gives a fairly complete generalization of the classical theory for smooth source and target spaces to their natural synthetic counterparts and an affirmative answer to a question raised several times in the recent literature.
Mondino et al. (Tue,) studied this question.