Multiply Harmonic Functions

Abstract

Let Ω and Ω′ be two locally compact, connected Hausdorff spaces having countable bases. On each of the spaces is defined a system of harmonic functions satisfying the axioms of M. Brelot [2]. The following is the description of such a system. To each open set of Ω is assigned a vector space of finite continuous functions, called the harmonic functions, on this set.

Removable singularities for 𝑛-harmonic functions and Hardy classes in polydiscs

“…A function is 2-harmonic on ß if it is continuous and separately harmonic. The 2-subharmonic and 2-superharmonic functions are defined similarly [4]. We remark that we consider 2 spaces only for the sake of convenience.…”

Integral representation for a class of multiply superharmonic functions

“…In 1965, the author showed that there is an unique integral representation for the subclass of (nS^) consisting of positive n-harmonic functions with the aid of Radon measures on the set of extreme elements belonging to a compact base, [5]. In 1968, R. Cairoli gave an unique integral representation for functions of two variables that are separately excessive and satisfy an additional condition, that is called condition (H) by him, [3].…”

Polydiscs and nontangential limits

“…This implies that the « -t.limhm/u is > min(/, m), pu a.e. Clearly the sequence of «-harmonic functions hm increases and by Harnack property [4] converges to a «-harmonic function h as m tends to oo and h < min^ , w2). Let now «' be the greatest «-harmonic minorant of min(wx, w2).…”

“…In terms of potential theoretical expectations it is natural to expect the result to extend to bounded quotients of positive «-harmonic functions. More precisely, suppose u > 0 is an «-harmonic function on D" that is represented as the integral of a finite Borel measure pu on T" relative to the product of the Poisson kernels [4,8]. Let w be any w-bounded (i.e., \w/u\ bounded) «-harmonic function on D" .…”