Distributions and measures on the boundary of a tree

In this paper, we analyze the space D of distributions on the boundary Ω of a tree and its subspace B0, 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 μ ∈ B0, then μ is a measure which is absolutely continuous with respect to the natural probability measure λ on Ω, but on the other hand there are measures absolutely continuous with respect to λ which are not in B0. 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 Ω 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 μ, Ω decomposes into two subspaces. On one of them, a union of intervals Aμ, μ restricted to any finite union of intervals extends to a complex measure and on Aμ 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 Aμ 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.  2004 Published by Elsevier Inc.