Lipschitz conditions on bounded harmonic functions on the upper half-space

This work is devoted to Lipschitz conditions on bounded harmonic functions on the upper half-space in Rn. Among other results, we prove the following one. Let U(x′,xn) be a real-valued bounded harmonic function on the upper half-space R+n={(x′,xn):x′∈Rn−1,xn∈(0,∞)}, which is continuous on the closure of this domain. Assume that for α∈(0,1), there exists a constant C such that for every x′∈Rn−1, we have ||U|(x′,xn)−|U|(x′,0)|≤Cxnα, xn∈(0,∞). Then there exists a constant C˜ such that |U(x)−U(y)|≤C˜|x−y|α,x,y∈R+n.