Kohur Gowrisankaran

Extreme harmonic functions and boundary value problems

The extreme elements of any base of the cone of positive harmonic functions on any open domain of R/ play an important role in the theory of harmonic functions. R. S. Martin [19], while generalising the Poisson-Stieltjes formula, gives an integral representation for non-negative harmonic functions with measures on the set of extreme elements of a base of the set of non-negative harmonic functions. Martin proves that all the extreme functions (and in fact some other functions too) form boundary elements of D in a suitable metric topology (which induces on D the topology of the euclidean space); this boundary is known after Martin as the Martin boundary () of D. M. Brelot [2, 3, 4, 5] continues the study of the (Martin) boundary and further extends the results to the case of Green spaces. He considers the Dirichlet problem (the first boundary value problem) on any Green space for the Martin boundary and moreover the relativized problem with the limits at the boundary of quotients of functions by a fixed positive harmonic function A. He demonstrates that for continuous functions on the boundary the solution by the Perron method exists, even in the case of the latter problem and for every A.

Measurability of functions in product spaces

Let f be a function on a product space Xx Y with values in a separable metrizable space such that it is measurable in one variable and continuous in the other. The joint measurability of such a function is proved under certain conditions on X and Y. Let X and Y be Hausdorff topological spaces. Let f be a complex valued function on the product space Xx Y such that f(, y) is Borel measurable on X for every ye Y and f (x, ) continuous on Y for every x in X. The problem of proving measurability off on Xx Y, as a function of the variables together is of interest and has occupied the attention of many mathematicians ([2], [3], [3bis], [5], [6]). For instance, in [2, p. 122], it is proved that if Yis a locally compact separable metric space (and Xany measure space, not necessarily a topological space) then f is jointly measurable. (See also [3].) In a recent paper, while considering the iterated fine limits of holomorphic functions [1], we had to prove the measurability of functions of the formf as above, but with values in a separable metric space where neither X nor Y is necessarily locally compact. Our proof in this case carries over to more general situations. We state below the main results. THEOREM 1. Let X and Y be Hausdorff topological spaces such that every compact subspace of Y is metrizable. Let it and v be any two Radon measures on X and Y respectively [4]. Let f: Xx Y-*A, where A is a separable metrizable space, be such that it is i-measurable on Xfor every yeY and continuous on Yfor every xeX. Then, f is It x v-Lusin measurable. THEOREM 2. Let (X, r) be a measurable space (i.e. a set X with a aalgebra of subsets of X); and Y a Suslin space. Let B be the (-algebra of all measurable subsets for a locally finite measure ,u on the Borel aalgebra of Y. Then, a function f on Xx Ywith values in a separable metrizable space A, -r-measurable for every yeY and continuous on Y for each xeX, is r x B-measurable on X x Y. Received by the editors September 15, 1970 and, in revised form, April 6, 1971. AMS 1969 subject classqifcations. Primary 2820.

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.

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.

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).

On a problem of Doob concerning multiply superharmonic functions

The following is a well-known result due to A.P. Calderon [2], asserting the existence of non-tangential limits of multiply harmonic functions.

Iterated fine limits

Let v and u be, respectively n-superharmonic and n-harmonic functions on the product of n harmonic spaces. We prove that the iterated fine limits of V exist and are independent of the order, for A almost every minimal U boundary element where A represents the function u. As an application we prove an important property concerning the reduced function of a positive nharmonic function.

Lusin and Suslin topologies on a set

It is proved that the Borel subsets of two Suslin topologies on a set X are identical if and only if there is a Suslin topology on X finer than both the given topologies. Properties of the supremum topology are given under suitable conditions on the subdivisions of the spaces. A Hausdorff topological space is said to be Lusin (resp. Suslin) if there is a continuous bijective mapping (resp. a continuous surjective mapping) from a complete separable metric space onto X. The topological spaces which are Suslin or Lusin have a number of interesting measure theoretic properties. We mention two of them here. The first and perhaps one of the very interesting properties is that if X is a Lusin (resp. Suslin) space and X is a subspace of a Hausdorff topological space Y, then X is Borel in Y (resp. X is measurable for every finite Borel measure on Y). The second property is that every finite Borel measure on a Suslin space is a Radon measure. The development of the Suslin spaces and, in particular, the proofs of the above results can be found in [1]. In this note we consider a set X with two Suslin topologies r, and -r2. It is of importance and interest to know when the r1-Borel sets and r2Borel sets are identical. It is known that this happens if the inf(Q1, r2) is a Hausdorff topology. We shall show that a necessary and sufficient condition for the above to happen is that sup(Q1, r2) is a Suslin topology (Theorem 1). We also prove some properties of the supremum topology in terms of conditions on subdivisions on the space. We first have the THEOREM 1. Let rl and r2 be two Suslin (resp. Lusin) topologies on a set X. The r1-Borel sets and r2-Borel sets are identical if and only if the SUp(Q1, r2) is a Suslin (resp. Lusin) topology on X. PROOF. Assume that sup(Q1, -r) is a Suslin topology on X. It is well known that the Borel sets of two comparable Suslin topologies on a set Presented to the Society, September 1, 1972; received by the editors July 13, 1972. AMS (MOS) subject classifications (1970). Primary 28A05; Secondary 54H05, 04A1 5.

Superharmonic Tangential Approximation on Long Islands

We prove uniform and tangential approximation theorems for superharmonic functions in abstract harmonic spaces. Our tangential approximation theorem differs from traditional ones by the absence of the so-called long islands condition. The result is new also in the case of the classical potential theory.