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 a t x , y be a nonnegative measurable function on X × X which integrates to 1 in each variable. For a function f ∈ L 2 X and t ∈ T , define A t f x ≡ ∫   a t x , y f y   d y . We assume that A t f converges to f in L 2 , as t ⟶ 0 in T . For example, A t is a diffusion semigroup (with T = 0 , ∞ ). For W a finite measure space and w ∈ W , select real-valued h w ∈ L 2 X , defined everywhere, with h w L 2 X ≤ 1 . Define the distance D by D x , y ≡ h w x − h w y L 2 W . Our main result is an equivalence between the smoothness of an L 2 X function f (as measured by an L 2 -Lipschitz condition involving a t · , · and the distance D ) and the rate of convergence of A t f to f .