An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family

Abstract

Let X be a topological space equipped with a complete positive σ-finite measure and T a subset of the reals with 0 as an accumulation point. Let atx,y be a nonnegative measurable function on X×X which integrates to 1 in each variable. For a function f∈L2X and t∈T, define Atfx≡∫ atx,yfy dy. We assume that Atf converges to f in L2, as t⟶0 in T. For example, At is a diffusion semigroup (with T=0,∞). For W a finite measure space and w∈W, select real-valued hw∈L2X, defined everywhere, with hwL2X≤1. Define the distance D by Dx,y≡hwx−hwyL2W. Our main result is an equivalence between the smoothness of an L2X function f (as measured by an L2-Lipschitz condition involving at·,· and the distance D) and the rate of convergence of Atf to f.