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

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 at(x, y) be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L2(X) and t ∈ T, define Atf(x) ≡ ∫ at(x, y)f(y) 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 ∈ L2(X), 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 L2(X) function f (as measured by an L2‐Lipschitz condition involving at(·, ·) and the distance D) and the rate of convergence of Atf to f.