Lipschitz conditions on the modulus of a harmonic function

Abstract

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.