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+epsilon. 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 Calderon-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.

The behavior of harmonic functions at singular points of RCD spaces / De Philippis, G; Núñez-Zimbrón, J. - In: MANUSCRIPTA MATHEMATICA. - ISSN 0025-2611. - 171:1-2(2023), pp. 155-168. [10.1007/s00229-021-01365-9]

The behavior of harmonic functions at singular points of RCD spaces

De Philippis, G;
2023-01-01

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+epsilon. 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 Calderon-Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.
2023
171
1-2
155
168
https://doi.org/10.1007/s00229-021-01365-
https://arxiv.org/abs/1909.05220
De Philippis, G; Núñez-Zimbrón, J
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/135453
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