Boundary behavior p-harmonic functions in the Heisenberg group

TL;DR

This paper investigates how nonnegative p-harmonic functions behave near the boundary of domains in the Heisenberg group, establishing estimates that describe their vanishing rates relative to the boundary distance.

Contribution

It provides new boundary behavior estimates for p-harmonic functions in the Heisenberg group, including linear vanishing bounds and exact boundary vanishing rates for smooth domains.

Findings

p-harmonic functions vanish at most linearly near the boundary

Exact vanishing rate matches the boundary distance in C^{1,1} domains away from characteristic set

Comparison theorem shows functions vanish at the same rate near the boundary

Abstract

We study the boundary behavior of nonnegative p-harmonic functions which vanish on a portion of the boundary of a domain in the Heisenberg group H^n. Our main results are: 1) An estimate from above which shows that, under suitable geometric assumptions on the relevant domain, such a p-harmonic function vanishes at most linearly with respect to the sub-Riemannian distance to the boundary. 2) An estimate from below which shows that for a (Euclidean) C^{1,1} domain, away from the characteristic set, such a p-harmonic function vanishes exactly like the distance to the boundary. By combining 1) and 2) we obtain a comparison theorem stating that, at least away from the characteristic set, any two such p-harmonic functions must vanish at the same rate.