Opuscula Math. 33, no. 4 (2013), 697-700

Opuscula Mathematica

A note on bounded harmonic functions over homogeneous trees

Francisco Javier González Vieli

Abstract. Let \(\mathcal{T}_k\) be the homogeneous tree of degree \(k\geq 3\). J.M. Cohen and F. Colonna have proved that if \(f\) is a bounded harmonic function on \(\mathcal{T}_k\), then \(|f(x)-f(y)|\leq \|f\|_\infty\cdot 2(k-2)/k\) for any adjacent vertices \(x\) and \(y\) in \(\mathcal{T}_k\). We give here a new and very simple proof of this inequality.

Keywords: bounded harmonic function, homogeneous tree.

Mathematics Subject Classification: 31C20, 05C05, 05C63.

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  • Francisco Javier González Vieli
  • Montoie 45, 1007 Lausanne, Switzerland
  • Received: 2013-04-22.
  • Accepted: 2013-05-22.
Opuscula Mathematica - cover

Cite this article as:
Francisco Javier González Vieli, A note on bounded harmonic functions over homogeneous trees, Opuscula Math. 33, no. 4 (2013), 697-700, http://dx.doi.org/10.7494/OpMath.2013.33.4.697

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