JournalsrmiVol. 23, No. 3pp. 831–845

Lipschitz conditions on the modulus of a harmonic function

  • Miroslav Pavlović

    University of Belgrade, Beograd, Serbia
Lipschitz conditions on the modulus of a harmonic function cover
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Abstract

It is proved that if uu is a real valued function harmonic in the open unit ball BNRN\mathbb B_N\subset \mathbb R^N and continuous on the closed ball, then the following conditions are equivalent, for 0<α<10 < \alpha < 1: \begin{itemize} \item u(x)u(w)Cxwα,x,wBN|u(x)-u(w)|\le C|x-w|^\alpha, \quad x, w\in \mathbb B_N; \item u(y)u(ζ)Cyζα,y,ζBN| |u(y)|-|u(\zeta) | |\le C|y-\zeta|^\alpha, \quad y, \zeta\in \partial\mathbb B_N; \item u(y)u(ry)C(1r)α,yBN, 0<r<1| |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 up|u|^p is also considered.

Cite this article

Miroslav Pavlović, Lipschitz conditions on the modulus of a harmonic function. Rev. Mat. Iberoam. 23 (2007), no. 3, pp. 831–845

DOI 10.4171/RMI/515