On a Theorem of Wolff Revisited

Abstract

We study $p$-harmonic functions, $ 1 < p\neq 2 < \infty$, in $ \mathbb{R}^{2}_+ = \{ z = x + i y : y > 0, - \infty < x < \infty \} $ and $B( 0, 1 ) = \{ z : |z| < 1 \}$. We first show for fixed $ p$, $1 < p\neq 2 < \infty$, and for all large integers $N\geq N_0$ that there exists $p$-harmonic function, $ V = V ( r e^{i\theta} )$, which is $ 2\pi/N $ periodic in the $ \theta $ variable, and Lipschitz continuous on $ \partial B (0, 1)$ with Lipschitz norm $\leq c N$ on $ \partial B ( 0, 1 )$ satisfying $V(0)=0$ and $ c^{-1} \leq \int_{-\pi}^{\pi} V ( e^{i\theta} ) d \theta \leq c$. In case $2<p<\infty $ we give a more or less explicit example of $V$ and our work is an extension of a result of Wolff on $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$. Using our first result, we extend the work of Wolff on failure of Fatou type theorems for $ \mathbb{R}^{2}_+ $ to $ B (0, 1)$ for $p$-harmonic functions, $1< p\neq 2<\infty$. Finally, we also outline the modifications needed for extending the work of Llorente, Manfredi, and Wu regarding failure of subadditivity of $p$-harmonic measure on $ \partial \mathbb{R}^{2}_+ $ to $\partial B (0, 1)$.