Phragmén-Lindelöf theorems and p-harmonic measures for sets near low-dimensional hyperplanes

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 Phragmén-Lindelöf 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.