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

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 $Λ\subset \mathbf{R}^n$, $m \in [0,n-1]$. Using these estimates, we derive results of Phragmén-Lindelöf type in unbounded domains $Ω\subset \mathbf{R}^n\setminus Λ$ 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.

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