Estimates of $p$-harmonic functions in planar sectors

Volume 61, Issue 1 (2023), pp. 141–175

Niklas L. P. Lundström

^{ }Jesper Singh^{ }
Pub. online: 26 April 2023
Type: Article

Received

2 November 2021

2 November 2021

Revised

1 July 2022

1 July 2022

Accepted

1 August 2022

1 August 2022

Published

26 April 2023

26 April 2023

#### Abstract

Suppose that $p \in (1,\infty], \nu \in [1/2,\infty), {S_{\nu }}=\left\{({x_{1}},{x_{2}})\in {\mathbb{R}^{2}}\setminus \{(0,0)\}:|\phi | \lt \frac{\pi }{2\nu }\right\}$, where $\phi$ is the polar angle of $(x_1, x_2)$. Let $R \gt 0$ and $\omega_p (x)$ be the $p$-harmonic measure of $\partial B(0,R) \cap \mathcal{S}_\nu$ at $x$ with respect to $B(0,R) \cap S_\nu$. We prove that there exists a constant $C$ such that

\[
C^{-1} {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)} \leq \omega_p (x) \leq C {\left( \dfrac{\lvert x \rvert}{R} \right)}^{k (\nu, p)}
\]

whenever $x \in B(0,R) \cap \mathcal{S}_{2 \nu}$ and where the exponent $k(\nu, p)$ is given explicitly as a function of $\nu$ 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 Phragmén–Lindelöf theorem for $p$-subharmonic functions in the unbounded sector $S_\nu$. 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 $\nu \in (1/2,\infty)$ and $p \in (1,\infty)$ we prove uniqueness (modulo normalization) of positive $p$-harmonic functions in $\mathcal{S}_\nu$ vanishing on $\partial \mathcal{S}_\nu$.