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 $\sigma$-finite measure and $T$ a subset of the reals with $0$ as an accumulation point. Let $a_t\left(x,y\right)$ be a nonnegative measurable function on $X\times X$ which integrates to $1$ in each variable. For a function $f\in L_2\left(X\right)$ and $t\in T$, define $A_{t}f\left(x\right)\equiv {\int }^{\text{}}{a}_t\left(x,y\right)f\left(y\right)dy$. We assume that $A_{t}f$ converges to $f$ in $L_2$, as $t\to 0$ in $T$. For example, $A_t$ is a diffusion semigroup $(withT=[0,\infty )).$ For $W$ a finite measure space and $w\in W$, select real-valued $h_w\in L_2\left(X\right)$, defined everywhere, with ${\Vert h_w\Vert }_{L_2(X)}\le 1$. Define the distance $D$ by $D\left(x,y\right)\equiv \Vert h_w\left(x\right)-h_w\left(y\right){\Vert }_{L_2\left(w\right)}$. Our main result is an equivalence between the smoothness of an $L_2\left(X\right)$ function $f$ (as measured by an $L_2$-Lipschitz condition involving $a_t\left(.,.\right)$and the distance $D$) and the rate of convergence of $A_{t}f$ to $f$.