Distributions and measures on the boundary of a tree

In this paper, we analyze the space D of distributions on the boundary Omega of a tree and its subspace B-0, which was introduced in [Amer. J. Math. 124 (2002) 999-1043] in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if mu is an element of B-0, then mu is a measure which is absolutely continuous with respect to the natural probability measure lambda on Omega, but on the other hand there are measures absolutely continuous with respect to lambda which are not in B-0. We then give the definition of an absolutely summable distribution and prove that a distribution can be extended to a complex measure on the Borel sets of Omega if and only if it is absolutely summable. This is also equivalent to the condition that the distribution have finite total variation. Finally, we show that for a distribution mu, Omega decomposes into two subspaces. On one of them, a union of intervals A(mu), mu restricted to any finite union of intervals extends to a complex measure and on A(mu) we give a version of the Jordan, Hahn, and Lebesgue-Radon-Nikodym decomposition theorems. We also show that there is no interval in the complement of A(mu) in which any type of decomposition theorem is possible. All the results in this article can be generalized to results on good (in particular, compact infinite) ultrametric spaces, for example, on the p-adic integers and the p-adic rationals. (C) 2004 Published by Elsevier Inc.