Estimates for Nonlinear Harmonic Measures on Trees

In this paper we give some estimates for nonlinear harmonic measures on trees. In particular, we estimate in terms of the size of a set $D$ the value at the origin of the solution to $u(x)=F((x,0),\dots,(x,m-1))$ for every $x\in\mathbb{T}_m,$ a directed tree with $m$ branches with initial datum $f+\chi_D$. Here $F$ is an averaging operator on $\mathbb{R}^m$, $x$ is a vertex of a directed tree $\mathbb{T}_m$ with regular $m$-branching and $(x,i)$ denotes a successor of that vertex for $0\le i\le m-1$.