## Journal of Differential Geometry

### Hölder Estimates and Regularity for Holomorphic and Harmonic Functions

#### Abstract

In this paper, we proved that if a singular manifold satisfies a weak mean value property for positive subharmonic functions then one can derive an oscillation bound for bounded holomorphic functions. Moreover, if we further assume that the volume decays at most polynomially at a singular point, then we obtain a Hölder estimate of the holomorphic function at that point. In a similar spirit, we also established a continuity estimate for bounded harmonic functions, with a finite dimensional exception, at a singular point of a manifold satisfying a weak mean value property and a polynolmial volume decay condition.

#### Article information

Source
J. Differential Geom., Volume 58, Number 2 (2001), 309-329.

Dates
First available in Project Euclid: 20 July 2004

https://projecteuclid.org/euclid.jdg/1090348328

Digital Object Identifier
doi:10.4310/jdg/1090348328

Mathematical Reviews number (MathSciNet)
MR1913945

Zentralblatt MATH identifier
1049.58024

#### Citation

Li, Peter; Wang, Jiaping. Hölder Estimates and Regularity for Holomorphic and Harmonic Functions. J. Differential Geom. 58 (2001), no. 2, 309--329. doi:10.4310/jdg/1090348328. https://projecteuclid.org/euclid.jdg/1090348328