Phragm\'en-Lindel\"of theorems and p-harmonic measures for sets near low-dimensional hyperplanes

TL;DR

This paper establishes estimates for p-harmonic measures near low-dimensional hyperplanes and derives Phragmén-Lindelöf type results and growth estimates for p-harmonic functions in such settings.

Contribution

It introduces new estimates for p-harmonic measures near hyperplanes and applies them to derive growth and boundary behavior results for p-harmonic functions.

Findings

Estimates of p-harmonic measure near hyperplanes

Phragmén-Lindelöf type theorems for p-subharmonic functions

Growth estimates for p-harmonic functions

Abstract

We prove estimates of a $p$-harmonic measure, $p \in (n-m, \infty]$, for sets in $\mathbf{R}^n$ which are close to an $m$-dimensional hyperplane $\Lambda \subset \mathbf{R}^n$, $m \in [0,n-1]$. Using these estimates, we derive results of Phragm\'en-Lindel\"of type in unbounded domains $\Omega \subset \mathbf{R}^n\setminus \Lambda$ for $p$-subharmonic functions. Moreover, we give local and global growth estimates for $p$-harmonic functions, vanishing on sets in $\mathbf{R}^n$, which are close to an $m$-dimensional hyperplane.