A condition equivalent to the Hölder continuity of harmonic functions on unbounded Lipschitz domains

Our main result concerns the behavior of bounded harmonic functions on a domain in [Formula: see text] which may be represented as a strict epigraph of a Lipschitz function on [Formula: see text]. Generally speaking, the result says that the Hölder continuity of a harmonic function on such a domain is equivalent to the uniform Hölder continuity along the straight lines determined by the vector [Formula: see text], where [Formula: see text] is the base of standard vectors in [Formula: see text]. More precisely, let [Formula: see text] be a Lipschitz function on [Formula: see text], and [Formula: see text] be a real-valued bounded harmonic function on [Formula: see text]. We show that for [Formula: see text] the following two conditions on [Formula: see text] are equivalent: (a) There exists a constant [Formula: see text] such that [Formula: see text] (b) There exists a constant [Formula: see text] such that [Formula: see text] Moreover, the constant [Formula: see text] depends linearly on [Formula: see text]. The result holds as well for vector-valued harmonic functions and, therefore, for analytic mappings.