Estimates for nonlinear harmonic measures on trees

LEANDRO DEL PEZZO, CAROLINA MOSQUERA, JULIO ROSSI

BULLETIN OF THE BRAZILIAN MATHEMATICAL SOCIETY

SPRINGER

Lugar: Berlin; Año: 2014

1678-7544

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 $xin T_m,$ a directed tree with m branches with initial datum $f+chi_D.$ Here $F$  is an averaging operator on $R^m,$ $x$ is a vertex of a directed tree $T_m$ with regular m branching and  $(x,i)$ denotes a successor of that vertex for $0le ile m-1.$