On a Theorem of Wolff Revisited

Abstract

We study $p$-harmonic functions, $1<p\ne 2<\infty$, in ${\mathbb{R}}_{+}^2=\{z=x+iy:y>0,-\infty <x<\infty \}$ and $B(0,1)=\{z:|z|<1\}$. We first show for fixed $p$, $1<p\ne 2<\infty$, and for all large integers $N\ge N_0$ 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 $\le cN$ on $\partial B(0,1)$ satisfying $V(0)=0$ and $c^{-1}\le {\int }_{-\pi }^{\pi }V(e^{i\theta })d\theta \le c$. In case $2<p<\infty$ 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, $1<p\ne 2<\infty$. 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 $\partial B(0,1)$.

38 pages, 1 figure