Carleson Measures for Non-negative Subharmonic Functions on Homogeneous Trees

Abstract

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 σ on a homogeneous tree T, equipped with the nearest-neighbor transition operator and studied their relationships under certain assumptions on σ. We defined two classes of measures σ we called good and optimal and showed that if σ is optimal and μ is a σ-Carleson measure on T in the sense that there is a constant C such that the μ measure of every sector is bounded by C times the σ measure of the sector, then there exists C_μ > 0 such that $\sum f(v) \mu (v) \le C_{\mu } \sum f(v) \sigma (v)$ for every non-negative subharmonic function f on T, and we conjectured that this holds if and only if σ 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.