Abstract. In this paper, we introduce and study polyharmonic functions on trees. We prove that the discrete version of the classical Riquier problem can be solved on trees. Next, we show that the discrete version of a characterization of harmonic functions due to Globevnik and Rudin holds for biharmonic functions on trees. Furthermore, on a homogeneous tree we characterize the polyharmonic functions in terms of integrals with respect to finitely-additive measures (distributions) of certain functions depending on the Poisson kernel. We define polymartingales on a homogeneous tree and show that the discrete version of a characterization of polyharmonic functions due to Almansi holds for polymartingales. We then show that on homogeneous trees there are 1-1 correspondences among the space of nth-order polyharmonic functions, the space of nth-order polymartingales, and the space of n-tuples of distributions. Finally, we study the boundary behavior of polyharmonic functions on homogeneous trees whose associated distributions satisfy various growth conditions.
In this paper, we analyze the space D of distributions on the boundary Ω of a tree and its subspace B 0 , which was introduced in [Amer. J. Math. 124 (2002Math. 124 ( ) 999-1043 in the homogeneous case for the purpose of studying the boundary behavior of polyharmonic functions. We show that if µ ∈ B 0 , 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 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 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 A µ , µ restricted to any finite union of intervals extends to a complex measure and on A µ 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 µ 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.
ABSTRACT. In this paper, a general Fatou theorem is obtained for functions which are integrals of kernels against measures on R n. These include solutions of Laplace's equation on an upper half-space, parabolic equations on an infinite slab and the heat equation on a right half-space. Lebesgue almost everywhere boundary limits are obtained within regions which contain sequences approaching the boundary with any prescribed degree of tangency.
The functions H and Hv, which first appear in [3, pp. 5–6], play a major role throughout this paper. One of the stated properties of Hv is that it is harmonic everywhere except at v. However, Hv fails to be harmonic at any terminal vertex. Accordingly, we assume that the tree has no terminal vertices. With this assumption, all stated properties and results of [3] are correct.
This error was first observed in connection with [1] (cf. [2]) where the functions Hv were first introduced.
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