The behavior of harmonic functions at singular points of $\mathsf{RCD}$ spaces

TL;DR

This paper studies the behavior of harmonic functions at singular points in RCD spaces, revealing conditions under which their gradients vanish and discussing implications for regularity and Calderón-Zygmund estimates.

Contribution

It provides new insights into the gradient behavior of harmonic functions at singular points in RCD spaces and explores limitations of regularity estimates based solely on curvature bounds.

Findings

Gradient vanishes at points with tangent cones over non-maximal diameter

No a priori gradient modulus estimate depending only on sectional curvature

No a priori Calderón-Zygmund theory based solely on curvature bounds

Abstract

In this note we investigate the behavior of harmonic functions at singular points of $\mathsf{RCD}(K,N)$ spaces. In particular we show that their gradient vanishes at all points where the tangent cone is isometric to a cone over a metric measure space with non-maximal diameter. The same analysis is performed for functions with a Laplacian in $L^{N+\varepsilon}$. As a consequence we show that on smooth manifolds there is no a priori estimate on the modulus of continuity of the gradient of harmonic functions which depends only on lower bounds of the sectional curvature. In the same way we show that there is no a priori Calder\'on-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.