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

We show how the $A_{\mathrm{\infty }}$ 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.