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 $0<\alpha <1$:

• $|u(x)-u(w)|\le C|x-w{|}^{\alpha },x,w\in {\mathbb{B}}_N$;
• $||u(y)|-|u(\zeta )||\le C|y-\zeta {|}^{\alpha },y,\zeta \in \partial {\mathbb{B}}_N$;
• $||u(y)|-|u(ry)||\le C(1-r{)}^{\alpha },y\in \partial {\mathbb{B}}_N,0<r<1$.

The Lipschitz condition on $|u{|}^p$ is also considered.