@article{1c8ea124cb4049d3ad37653b5677a4ce,

title = "The behavior of harmonic functions at singular points of RCD spaces",

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 Calder{\'o}n–Zygmund theory for the Laplacian with bounds depending only on lower bounds of the sectional curvature.",

keywords = "Calder{\'o}n-Zygmund theory, RCD space, harmonic function",

author = "{De Philippis}, Guido and Jes{\'u}s N{\'u}{\~n}ez-Zimbr{\'o}n",

note = "Funding Information: 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. Funding Information: 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. Publisher Copyright: {\textcopyright} 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.",

year = "2023",

month = may,

doi = "10.1007/s00229-021-01365-9",

language = "English (US)",

volume = "171",

pages = "155--168",

journal = "manuscripta mathematica",

issn = "0025-2611",

publisher = "Springer New York",

number = "1-2",

}