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

created by dephilipp on 12 Sep 2019

[BibTeX]

preprint

Inserted: 12 sep 2019
Last Updated: 12 sep 2019

Year: 2019

ArXiv: 1909.05220 PDF

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.