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

We show how the $A_{\infty}$ class of weights can beconsidered as a metric space. As far as we know this is thefirst time that a metric $d_{*}$ is considered on this set.We use this metric to generalize the results obtained in\cite{NV}. Namely, we show that for any Calderón-Zygmundoperator $T$ and an $A_{p}$, $1<p<\infty$, weight $w_{0}$,the numbers $\|T\|_{L^{p}(w)\rightarrow L^{p}(w)}$ convergeto $\|T\|_{L^{p}(w_{0})\rightarrow L^{p}(w_{0})}$ as$d_{*}(w,w_{0})\to 0$. We also find the rate of thisconvergence and prove that it is sharp.

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Published 12 July 2012