On a Theorem of Wolff Revisited

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 $\partial B (0, 1)$.