An extrapolation theorem with applications to weighted estimates for singular integrals

We prove an extrapolation theorem saying that the weighted weak type (1,1)(1,1) inequality for A1A1 weights implies the strong Lp(w)Lp(w) bound in terms of the Lp(w)Lp(w) operator norm of the maximal operator M. The weak Muckenhoupt–Wheeden conjecture along with this result allows us to conjecture that the following estimate holds for a Calderón–Zygmund operator T   for any p>1p>1:‖T‖Lp(w)⩽c‖M‖Lp(w)p. The latter conjecture would yield the sharp estimates for ‖T‖Lp(w)‖T‖Lp(w) in terms of the AqAq characteristic of w   for any 1