A Difference Property for Functions with Bounded Second Differences on Amenable Topological Groups

Abstract

Let G be a topological group. For a function f : G → ℝ and h ∈ G, the right difference function Δ_hf is defined by Δ_hf(g) = f(gh) − f(g) (g ∈ G). A function H: G → ℝ is said to be additive if it satisfies the Cauchy functional equation H(g + h) = H(g) + H(h) for every g, h ∈ G. A class F of real-valued functions defined on G is said to have the difference property if, for every function f : G → ℝ satisfying Δ_hf ∈ F for every h ∈ G, there is an additive function H such that f − H ∈ F. The Erdős conjecture claiming that the class of continuous functions on ℝ has the difference property was proved by de Bruijn; later on, Carroll and Koehl proved a similar result for the compact Abelian groups and, under an additional assumption, for the compact metric groups, namely, under the assumption that all functions of the form ∇_hf(g) = f(hg)−f(g), g ∈ G, are Haar measurable for every h ∈ G. One of the consequences of this assumption is the boundedness of the function {g, h} ⟼ f(gh) − f(g) − f(h) + f(e), g, h ∈ G, for every function f on a compact group G for which the difference functions Δ_hf are continuous for every h ∈ G and the functions ∇_hf are Haar measurable for every h ∈ G (e stands for the identity element of the group G). In the present paper, we consider the difference property under the very strong assumption that the function {g, h} ⟼ f(gh) − f(g) − f(h) + f(e), g, h ∈ G, is bounded. This assumption enables us to obtain results concerning difference properties not only for functions on groups but also for functions on homogeneous spaces.