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 double script B signN ⊂ ℝN and continuous on the closed ball, then the following conditions are equivalent, for 0 < α < 1: |u(x) - u(w)| ≤ C|x - w|α, x, w ∈ double script B signN; ||u(y)| - |u(ζ)|| < C|y - ζ| α, y, ζ ∈ ∂double script B signN; ||u(y)| - |u(ry)|| ≤ C(1 - r)α, y ∈ ∂double script B signN, 0 < r < 1. The Lipschitz condition on |u|p is also considered.