The behavior of harmonic functions at singular points of RCD spaces

Guido De Philippis, Jesús Núñez-Zimbrón

Research output: Contribution to journalArticlepeer-review

Abstract

In this note we investigate the behavior of harmonic functions at singular points of 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 Laplacian in LN+ε. 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 Calderón–Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.

Original languageEnglish (US)
Journalmanuscripta mathematica
DOIs
StateAccepted/In press - 2022

Keywords

  • Calderón-Zygmund theory
  • RCD space
  • harmonic function

ASJC Scopus subject areas

  • Mathematics(all)

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