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

We prove estimates of a $p$-harmonic measure, $p\in (n-m,\mathrm{\infty }]$, 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.