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

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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 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 Calderón–Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.

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Notes

  1. Recall that a modulus of continuity is a continuous and bounded function \(\omega : (0,+\infty )\rightarrow (0,1]\) such that \(\omega (0+)=0\).

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Acknowledgements

We thank Nicola Gigli for encouraging us to write this note and for several useful discussions. G.D.P. is supported by an INDAM Grant “Geometric Variational Problems”. J.N.-Z. was supported by a MIUR SIR-Grant “Nonsmooth Differential Geometry” (RBSI147UG4) and a DGAPA-UNAM Postdoctoral Fellowship. J.N.-Z. also acknowledges support from CONACyT Project CB2016-283988-F.

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Authors and Affiliations

  1. SISSA, Via Bonomea 265, 34136, Trieste, Italy

    Guido De Philippis

  2. Centro de Ciencias Matemáticas UNAM, Antigua Carretera a Pátzcuaro 8701, 58089, Morelia, Michoacán, Mexico

    Jesús Núñez-Zimbrón

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  1. Guido De Philippis
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Correspondence to Jesús Núñez-Zimbrón.

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De Philippis, G., Núñez-Zimbrón, J. The behavior of harmonic functions at singular points of \(\mathsf {RCD}\) spaces. manuscripta math. 171, 155–168 (2023). https://doi.org/10.1007/s00229-021-01365-9

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