Calderón weights as Muckenhoupt weights

Abstract

The Calderón operator S is the sum of the the Hardy averaging operator and its adjoint. The weights w for which S is bounded on L p(w) are the Calderón weights of the class Cp. We prove a characterization of the weights in Cp by a single condition which allows us to see that Cp is the class of Muckenhoupt weights associated with a maximal operator defined through a basis in (0,∞). The same condition characterizes the weighted weak-type inequalities for 1 < p < ∞, but that the weights for the strong type and the weak type differ for p = 1. We also prove that the weights in Cp do not behave like the usual Ap weights with respect to some properties and, in particular, we answer an open question on extrapolation for Muckenhoupt bases without the openness property.