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 Jn. Among other results, we prove the following one. Let U.x'; xn / be a real-valued bounded harmonic function on the upper half-space Jn C = { .x ' ; xn/ W x' E Jn1;xn E .0;oo/}, which is continuous on the closure of this domain. Assume that for E .0; 1/, there exists a constant C such that for every x' E Jn_1, we have U .x'; xn/- U .x'; 0/ <= Cxn, xnE .0;oo/. Then there exists a constant CQ such that U.x/- U.y/ <= CQx- y;x;yEJnC.