David Singman

POLYHARMONIC FUNCTIONS ON TREES

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.

Function Series, Catalan Numbers, and Random Walks on Trees

The delight of finding unexpected connections is one of the rewards of studying mathematics. In this talk, based on joint work with Ibtesam Bajunaid, Joel Cohen, and David Singman, I will discuss the connections that link the following seven superficially unrelated entities: (A) A function of the sort that calculus textbooks often use to show that a continuous function need not have a derivative at each point:

A global Riesz decomposition theorem on trees without positive potentials

We study the potential theory of trees with nearest-neighbor transition probability that yields a recurrent random walk and show that, although such trees have no positive potentials, many of the standard results of potential theory can be transferred to this setting. We accomplish this by defining a non-negative function H, harmonic outside the root e and vanishing only at e, and a substitute notion of potential which we call H-potential. We define the flux of a superharmonic function outside a finite set of vertices, give some simple formulas for calculating the flux and derive a global Riesz decomposition theorem for superharmonic functions with a harmonic minorant outside a finite set. We discuss the connection of the H-potentials with other notions of potentials for recurrent Markov chains in the literature

Exceptional sets in a product of harmonic spaces

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Tangential Limits of Potentials on Homogeneous Trees

Let T be a homogeneous tree of homogeneity q+1. Let Δ denote the boundary of T, consisting of all infinite geodesics b=[b0,b1,b2,] beginning at the root, 0. For each bεΔ, τ≥1, and a≥0 we define the approach region Ωτ,a(b) to be the set of all vertices t such that, for some j, t is a descendant of bj and the geodesic distance of t to bj is at most (τ−1)j+a. If τ>1, we view these as tangential approach regions to b with degree of tangency τ. We consider potentials Gf on T for which the Riesz mass f satisfies the growth condition ∑Tfp(t)q−γ|t|<∞, where p>1 and 0<γ<1, or p=1 and 0<γ≤1. For 1≤τ≤1/γ, we show that Gf(s) has limit zero as s approaches a boundary point b within Ωτ,a(b) except for a subset E of Δ of τγ-dimensional Hausdorff measure 0, where Hτγ(E)=sup δ>0inf ∑iq−τγ|ti|:E a subset of the boundary points passing through ti for some i,|ti|>log q(1/δ).

Minimal Fine Limits for a Class of Potentials

We consider potentials Gk μ associated with the Weinstein equation with parameter k in ℝ, Σj=1n (∂2u/∂ x2j) + (k/xn) (∂ u/∂ xn) = 0, on the upper half space in ℝn. We show that if the representing measure μ satisfies the growth condition ∫ ynω/(1+|y|)n-k < ∞, where max(k, 2 − n) < ω ≤ 1, then Gk μ has a minimal fine limit of 0 at every boundary point except for a subset of vanishing (n − 2 + ω) dimensional Hausdorff measure. We also prove this exceptional set is best possible.

Generalized local Fatou theorems and area integrals

Let X be a space of homogeneous type and W a subset of X x (0, oo). Then, under minimal conditions on W, we obtain a relationship between two modes of convergence at the boundary X for functions defined on W. This result gives new local Fatou theorems of the Carleson-type for solutions of Laplace, parabolic and Laplace-Beltrami equations as immediate consequences of the classical results. Lusin area integral characterizations for the existence of limits within these more general approach regions are also obtained.

Boundary behavior of invariant Green’s potentials on the unit ball in ⁿ

Let p(z) = fB G(z, w) dA(w) be an invariant Greens potential on the unit ball B in Cn (n > 1), where G is the invariant Greens function and t is a positive measure with fB(1 Iw12)n dct(w) 0, is bounded away from 0. The result obtained here generalizes Lueckings result, see [L], on the unit disc in C.

Thin Sets and Boundary Behavior of Solutions of the Helmholtz Equation

The Martin boundary for positive solutions of the Helmholtz equation in n-dimensional Euclidean space may be identified with the unit sphere. Let v denote the solution that is represented by Lebesgue surface measure on the sphere. We define a notion of thin set at the boundary and prove that for each positive solution of the Helmholtz equation, u, there is a thin set such that u/v has a limit at Lebesgue almost every point of the sphere if boundary points are approached with respect to the Martin topology outside this thin set. We deduce a limit result for u/v in the spirit of Nagel–Stein (1984).