Estimates for Nonlinear Harmonic Measures on Trees

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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),,(x,m-1)) for every xₘ, a directed tree with m branches with initial datum f+χD. Here F is an averaging operator on Rᵐ, x is a vertex of a directed tree Tₘ with regular m-branching and (x,i) denotes a successor of that vertex for 0≤ i≤ m-1.