Harmonic functions on topological groups and symmetric spaces

Abstract

Let G be a metric group, not necessarily locally compact, acting on a metric space X , for instance, a right coset space of G. We introduce and develop a basic structure theory for harmonic functions on X which is applicable to infinite dimensional Riemannian symmetric spaces.

Harmonic functions on hypergroups

“…It is well known that Riemannian symmetric spaces can be represented as homogeneous spaces G/K of Lie groups G. Recently, harmonic functions on Riemannian symmetric spaces have been studied in [8,10] via convolution semigroups of measures on G, where the harmonic spherical functions identify as functions on the double coset space G//K. Since double coset spaces are special examples of hypergroups on which the Borel measures have a convolution structure, it is natural to consider harmonic functions on the wider class of hypergroups where convolution can be exploited.…”

Amenability, Reiter's condition and Liouville property

“…Proof. The arguments are similar to those given in [4] and [6] for groups, but we include the proof for completeness. Let LUC(S) be equipped with the topology ρ of pointwise convergence and let the Cartesian product LUC(S) LU C(S) be equipped with the product topology.…”

“…For f ∈ F m−1 and ρ ∈ Θ m−1 , set φ f,ρ (η) = G v γ (η)1 Am (γ)f (t(η))dλ t(η) (γ). Again, φ f,ρ is Borel because ρ is λ-adapted, and φ f,ρ lies in L 1 (G, µ ⋆ λ) as (6) and 7imply…”

“…This will follow readily from the fact that the space H π (S) of bounded π-harmonic functions on S is the range of a contractive projection on LUC(S), which commutes with left translations. To show this, we need the following two lemmas which are straightforward extension of similar results in [6] for groups.…”

Аналоги Свойства Лиувилля Для Гармонических Функций На Неограниченных Областях

“…Многие авторы исследовали свойство Лиувилля для решений некоторых дифференциальных уравнений и неравенств на различных пространствах (см. [7][8][9][10][11] и библиографию в них).…”