Growth of harmonic functions on biregular trees

Year 2022, Volume: 9 Issue: 2, 1 - 8, 13.05.2022

Francisco Javier Gonzalez Vieli

https://doi.org/10.13069/jacodesmath.1056555

Abstract

On a biregular tree of degrees $q+1$ and $r+1$, we study the growth of two classes of harmonic functions. First, we prove that if $f$ is a bounded harmonic function on the tree and $x$, $y$ are two adjacent vertices, then $|f(x)-f(y)|\leq 2 (qr-1)\|f\|_\infty/((q+1)(r+1))$, thus generalizing a result of Cohen and Colonna for regular trees. Next, we prove that if $f$ is a positive harmonic function on the tree and $x$, $y$ are two vertices with $d(x,y)=2$, then $f(x)/(qr)\leq f(y)\leq qr\cdot f(x)$.

Keywords

Biregular tree, Growth, Harmonic function

References

• [1] V. Anandam, Harmonic functions and potentials on finite and infinite networks, Springer, Heidelberg, Bologna (2011).
• [2] S. Axler, P. Bourdon, W. Ramey, Harmonic function theory, Springer-Verlag, New York (2001).
• [3] N. L. Biggs, Discrete mathematics, Clarendon Press, Oxford University Press, New York (1985).
• [4] P. Cartier, Fonctions harmoniques sur un arbre, Sympos. Math. 9 (1972) 203–270.
• [5] J. M. Cohen, F. Colonna, The Bloch space of a homogeneous tree, Bol. Soc. Mat. Mex. 37 (1992) 63–82.
• [6] E. Nelson, A proof of Liouville’s theorem, Proc. Amer. Math. Soc. 12(6) (1961) 995.
• [7] W. Woess, Random walks on infinite graphs and groups, Cambridge University Press (2000).