Moments of Riesz measures on Poincaré disk and homogeneous tree–A comparative study

One of the purposes of this paper is to clarify the strong analogy between potential theory on the open unit disk and the homogeneous tree, to which we dedicate an introductory section. We then exemplify this analogy by a study of Riesz measures. Starting from interesting work by Favorov and Golinskii (2009), we consider subharmonic functions on the open unit disk, resp. on the homogeneous tree. Supposing that we can control the way how those functions may tend to infinity at the boundary, we derive moment type conditions for the Riesz measures. On one hand, we generalise the previous results of Favorov and Golinskii (2009) for the disk, and on the other hand, we show how to obtain analogous results in the discrete setting of the tree.