2020 An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family
Maxim J. Goldberg, Seonja Kim
Abstr. Appl. Anal. 2020: 1-5 (2020). DOI: 10.1155/2020/8866826

## 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 ${A_t}f\left( x \right) \equiv \mathop \smallint \nolimits^ {a_t}\left( {x,y} \right)f\left( y \right)dy$. 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 $||{h_w}{||_{{L_2}(X)}} \le 1$. 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 $f$.

## Acknowledgments

As for the last many years, we are grateful to Raphy Coifman for his continued willingness to discuss mathematics with us. The first author was partially supported by sabbatical and Faculty Development Funding from Ramapo College of New Jersey.

## Citation

Maxim J. Goldberg. Seonja Kim. "An Equivalence between the Limit Smoothness and the Rate of Convergence for a General Contraction Operator Family." Abstr. Appl. Anal. 2020 1 - 5, 2020. https://doi.org/10.1155/2020/8866826

## Information

Received: 25 August 2020; Accepted: 17 October 2020; Published: 2020
First available in Project Euclid: 28 July 2020

Digital Object Identifier: 10.1155/2020/8866826