Carleson and Vanishing Carleson Measures on Radial Trees

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We extend a discrete version of an extension of Carleson’s theorem proved in [5] to a large class of trees that have certain radial properties. We introduce the geometric notion of s-vanishing Carleson measure on such a tree T (with s ≥ 1) and give several characterizations of such measures. Given a measure σ on T and p ≥ 1, let Lp(σ) denote the space of functions g defined on T such that |g|p is integrable with respect to σ and let Lp(∂T) be the space of functions f defined on the boundary of T such that |f|p is integrable with respect to the representing measure of the harmonic function 1.We prove the following extension of the discrete version of a classical theorem in the unit disk proved by Power.