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Decembar, 2007 Lipschitz conditions on the modulus of a harmonic function
Miroslav Pavlović
Rev. Mat. Iberoamericana 23(3): 831-845 (Decembar, 2007).


It is proved that if $u$ is a real valued function harmonic in the open unit ball $\mathbb B_N\subset \mathbb R^N$ and continuous on the closed ball, then the following conditions are equivalent, for $0 < \alpha < 1$: \begin{itemize} \item $|u(x)-u(w)|\le C|x-w|^\alpha, \quad x, w\in \mathbb B_N$; \item $| |u(y)|-|u(\zeta) | |\le C|y-\zeta|^\alpha, \quad y, \zeta\in \partial\mathbb B_N$; \item $| |u(y)|-|u(ry)| |\le C(1-r)^\alpha, \quad y\in \partial\mathbb B_N,\ 0 < r < 1$. \end{itemize} The Lipschitz condition on $|u|^p$ is also considered.


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Miroslav Pavlović. "Lipschitz conditions on the modulus of a harmonic function." Rev. Mat. Iberoamericana 23 (3) 831 - 845, Decembar, 2007.


Published: Decembar, 2007
First available in Project Euclid: 27 February 2008

zbMATH: 1148.31003
MathSciNet: MR2414494

Primary: 26A16 , 30A05 , 30B05

Keywords: Harmonic functions , Lipschitz condition

Rights: Copyright © 2007 Departamento de Matemáticas, Universidad Autónoma de Madrid

Vol.23 • No. 3 • Decembar, 2007
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