Transversally Lipschitz harmonic functions are Lipschitz

Let Ω ⊂ ℝ n be a bounded domain with C ∞ boundary. We show that a harmonic function in Ω that is Lipschitz along a family of curves transversal to bΩ is Lipschitz in Ω. The space of Lipschitz functions which we consider is defined using the notion of a majorant which is a certain generalization of the power functions t α, 0 < α < 1