Bergman Spaces and Carleson Measures on Homogeneous Isotropic Trees

Hastings studied Carleson measures for non-negative subharmonic functions on the polydisk and characterized them by a certain geometric condition relative to Lebesgue measure $\s$. Cima \& Wogen and Luecking proved analogous results for weighted Bergman spaces on the unit ball and other open subsets of $\mathC^n$. We consider a similar problem on a homogeneous tree, and study how the characterization and properties of Carleson measures for various function spaces depend on the choice of reference measure $\sigma$.