Continuity of weighted estimates for sublinear operators

Abstract

In this note we prove that if a sublinear operator T satisfies a certain weighted estimate in the $L^{p}(w)$ space for all $w\in A_{p}$, $1<p<+\infty$, then the operator norm of T on $L^{p}(w)$ is a continuous function of the weight $w$, with respect to a certain metric $d_{*}$ on $A_{p}$. This, generalizes a previous result on the same subject for linear operators.