On a Theorem of Wolff Revisited

We study $p$-harmonic functions, $1<p\mathrm{\ne }2<\mathrm{\infty }$, in ${\mathbb{R}}_{+}^2=\{z=x+iy:y>0,-\mathrm{\infty }<x<\mathrm{\infty }\}$ and $B( 0, 1 ) = { z : |z| < 1 }$\mathrm{.}Wefirstshowforfixed$ p$,$1 < p\neq 2 < \infty$,andforalllargeintegers$N\geq N_0$thatthereexists$p$-harmonic function, $V=V(r{e}^{i\theta })$, which is $2\pi \mathrm{/}N$ periodic in the $\theta$ variable, and Lipschitz continuous on $\mathrm{\partial }B(0,1)$ with Lipschitz norm $\le cN$ on $\mathrm{\partial }B(0,1)$ satisfying $V(0)=0$ and $c^{-1}\le {\int }_{-\pi }^{\pi }V(e^{i\theta })d\theta \le c$. In case $2<p<\mathrm{\infty }$ 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 $\mathrm{\partial }{\mathbb{R}}_{+}^2$ to $\mathrm{\partial }B(0,1)$.