On a Theorem of Wolff Revisited

We study p-harmonic functions, 1 < p ≠ 2 < ∞, in ℝ+2 = {z = x + iy: y > 0, −∞ < x < ∞} and B(0, 1) = {z: ∣z∣ < 1}. We first show for fixed p,1 < p ≠ 2 < ∞, and for all large integers N ≥ N0 that there exists a p-harmonic function on B(0, 1), V = V(reiθ), which is 2π/N periodic in the θ variable, and Lipschitz continuous on ∂B(0, 1) with Lipschitz norm ≤ cN, satisfying V(0) = 0 and c−1≤∫−ππV(eiθ)dθ≤c. In case 2 < p < ∞ we give a more or less explicit example of V and our work is an extension of a result of Wolff in [Wol07, Lemma 1] on ℝ+2 to B(0, 1). Using our first result, we extend the work of Wolff in [Wol07] on the failure of Fatou type theorems for ℝ+2 to B(0, 1) for p-harmonic functions, 1 < p ≠ 2 < ∞. Finally, we also outline the modifications needed for extending the work of Llorente, Manfredi, and Wu in [LMW05] regarding the failure of subadditivity of p-harmonic measure on ∂ℝ+2 to ∂B(0, 1).