Growth of harmonic functions on biregular trees

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

Abstract

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).

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Published
2022-04-29
Section
Articles