Estimates of $p$-harmonic functions in planar sectors

Abstract

Suppose that $p \in (1,\infty]$, $v \in [1/2,\infty)$ and $\mathcal{S}_v = \left\{ (x, y) \in \mathbb{R}^2 \setminus \{(0, 0)\}; |\phi| < \frac{\pi}{2v}\right\}$, where $\phi$ is the polar angle of $(x,y)$. Let $R>0$ and $\omega_p(x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_v$ at $x$ with respect to $B(0, R)\cap \mathcal{S}_v$. We prove that there exists a constant $C$ such that \begin{align*} C^{-1}\left(\frac{|x|}{R}\right)^{k(v,p)} \, \leq \omega_p(x) \, \leq C \left(\frac{|x|}{R}\right)^{k(v,p)} \end{align*} where the exponent $k(v,p)$ is given explicitly as a function of $v$ and $p$. Using this estimate we derive local growth estimates for $p$-sub- and $p$-superharmonic functions in planar domains which are locally approximable by sectors, e.g., we conclude bounds of the rate of convergence near the boundary where the domain has an inwardly or outwardly pointed cusp. Using the estimates of $p$-harmonic measure we also derive a sharp Phragmen-Lindel\of theorem for $p$-subharmonic functions in the unbounded sector $\mathcal{S}_v$. Moreover, if $p = \infty$ then the above mentioned estimates extend from the setting of two-dimensional sectors to cones in $\mathbb{R}^n$. Finally, when $v \in (1/2, \infty)$ and $p\in (1,\infty)$ we prove a uniqueness result (modulo normalization) for positive $p$-harmonic functions in $\mathcal{S}_v$ vanishing on $\partial\mathcal{S}_v$.