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 $A_{t}f\left(x\right)\equiv \int \hspace{0.17em}{a}_t\left(x,y\right)f\left(y\right)dy$ . We assume that $A_{t}f$ converges to $f$ in $L_2$ , as $t⟶0$ in $T$ . For example, $A_t$ is a diffusion semigroup (with $T=\left[0,\infty \right)$ ). For $W$ a finite measure space and $w\in W$ , select real-valued $h_w\in L_2\left(X\right)$ , defined everywhere, with ${‖h_w‖}_{L_2\left(X\right)}\le 1$ . Define the distance $D$ by $D\left(x,y\right)\equiv {‖h_w\left(x\right)-h_w\left(y\right)‖}_{L_2\left(W\right)}$ . 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 $f$ .