Carleson Measures for Non-negative Subharmonic Functions on Homogeneous Trees

In Cohen et al. (Potential Anal. 44(4), 745-766, 2016), we introduced several classes of Carleson-type measures with respect to a radial reference measure sigma on a homogeneous tree T, equipped with the nearest-neighbor transition operator and studied their relationships under certain assumptions on sigma. We defined two classes of measures sigma we called good and optimal and showed that if sigma is optimal and mu is a sigma-Carleson measure on T in the sense that there is a constant C such that the mu measure of every sector is bounded by C times the sigma measure of the sector, then there exists C-mu > 0 such that n-ary sumation f(v)mu(v)<= C mu n-ary sumation f(v)sigma(v) for every non-negative subharmonic function f on T, and we conjectured that this holds if and only if sigma is good. In this paper we develop tools for studying the above conjecture and identify conditions on a class of non-negative subharmonic functions for which we can prove the conjecture for all functions in such a class. We show that these conditions hold for the set of all non-negative subharmonic functions which are generated by eigenfunctions of the Laplacian on T.