An extrapolation theorem with applications to weighted estimates for singular integrals

Abstract

We prove an extrapolation theorem saying that the weighted weak type (1; 1) inequality for A1 weights implies the strong Lp(w) bound in terms of the Lp(w) operator norm of the maximal operator M. The weak Muchkenhoupt-Wheeden conjecture along with this result allows us to conjecture that the following estimate holds for a Calder´on-Zygmund operator T for any p > 1: ∥T∥ Lp(w) ≤ c∥M∥p Lp(w): The latter conjecture would yield the sharp estimates for ∥T∥ Lp(w) in terms of the Aq characteristic of w for any 1 < q < p. In this paper we get a weaker inequality ∥T∥ Lp(w) ≤ c∥M∥p Lp(w) log(1 + ∥M∥ Lp(w)) with the corresponding estimates for ∥w∥Aq when 1 < q < p.