Lipschitz regularity of Sobolev maps from $\mathrm{RCD}(K,N)$ spaces to $\mathrm{CAT}(0)$ spaces

Abstract

We establish a quantitative criterion for local Lipschitz regularity of Sobolev maps in the sense of Korevaar and Schoen from finite-dimensional \RCD(K,N) spaces into complete \CAT(0) spaces, without assuming harmonicity. We quantify the failure of local energy minimality by the harmonic replacement potential, a dyadic sum of the square roots of normalized energy gaps between a map and its harmonic replacements. We prove that this potential, together with an averaged energy term, controls the minimal weak upper gradient pointwise, with constants depending only on the curvature-dimension bounds and the scale. In particular, local boundedness of the replacement potential implies that the map admits a locally Lipschitz representative.