The Muckenhoupt $A_\infty$ class as a metric space and continuity of weighted estimates

TL;DR

This paper introduces a metric space structure on the $A_ abla$ class of weights, enabling the analysis of the continuity and convergence of weighted operator norms for Calderon-Zygmund operators.

Contribution

It is the first to define a metric on the $A_ abla$ class and uses it to establish the continuity and sharp convergence rates of operator norms.

Findings

The $A_ abla$ class can be equipped with a metric space structure.

Operator norms of Calderon-Zygmund operators are continuous with respect to this metric.

The convergence rate of the operator norms is proven to be sharp.

Abstract

We show how the $A_\infty$ class of weights can be considered as a metric space. As far as we know this is the first time that a metric d is considered on this set. We use this metric to generalize the results obtained in [9]. Namely, we show that for any Calderon- Zygmund operator T and an $A_p$, 1 < p < 1, weight $w_0$, the operator norm of T in $L^{p}(w)$ converge to the operator norm of T in L^{p}(w_{0})$ as d(w;w_0) goes to 0. We also find the rate of this convergence and prove that is sharp.