Growth of harmonic functions on biregular trees

  • Francisco Javier Gonzalez
Keywords: Biregular tree,Growth,Harmonic function


On a biregular tree of degrees q+1q+1 and r+1r+1, we study the growth of two classes of harmonic functions. First, we prove that if ff is a bounded harmonic function on the tree and xx, yy are two adjacent vertices, then |f(x)−f(y)|≤2(qr−1)∥f∥∞/((q+1)(r+1))|f(x)−f(y)|≤2(qr−1)‖f‖∞/((q+1)(r+1)), thus generalizing a result of Cohen and Colonna for regular trees. Next, we prove that if ff is a positive harmonic function on the tree and xx, yy are two vertices with d(x,y)=2d(x,y)=2, then f(x)/(qr)≤f(y)≤qr⋅f(x)f(x)/(qr)≤f(y)≤qr⋅f(x).


V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).

S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).

N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).

P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.

J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.

E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.

W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).