Continuity of weighted estimates for sublinear operators

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