Abstract :
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.
