TY - JOUR

T1 - On a Theorem of Wolff Revisited

AU - Akman, Murat

AU - Lewis, John

AU - Vogel, Andrew

N1 - Publisher Copyright:
© 2022, The Hebrew University of Jerusalem.

PY - 2022/8

Y1 - 2022/8

N2 - 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).

AB - 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).

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U2 - 10.1007/s11854-022-0200-0

DO - 10.1007/s11854-022-0200-0

M3 - Article

AN - SCOPUS:85131298540

SN - 0021-7670

VL - 146

SP - 487

EP - 530

JO - Journal d'Analyse Mathematique

JF - Journal d'Analyse Mathematique

IS - 2

ER -