An extrapolation theorem with applications to weighted estimates for singular integrals

We prove an extrapolation theorem saying that the weighted weak type (1, 1) inequality for A ₁ weights implies the strong L ^p(w) bound in terms of the L ^p(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>1: The latter conjecture would yield the sharp estimates for ||T||L ^p(w) in terms of the A _q characteristic of w for any 1<q<p. In this paper we get a weaker inequality with the corresponding estimates for ||w||A _q when 1<q<p.