Moments of Riesz measures on Poincar\'e disk and homogeneous tree -- a comparative study

Abstract

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 [A Blaschke-type condition for analytic and subharmonic functions and application to contraction operators. Linear and complex analysis, pp. 37-47, Amer. Math. Soc. Transl. (2) 226, Amer. Math. Soc., Providence, RI, 2009], we consider subharmonic functions on the open unit disk, resp. on the homogenous 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. One one hand, we generalise the previous results for the disk, and on the other hand, we show how to obtain analogous results in the discrete setting of the tree.