Calderón weights as Muckenhoupt weights

JAVIER DUOANDIKOEXTEA ; F.J. MARTÍN-REYES ; SHELDY OMBROSI

INDIANA UNIVERSITY MATHEMATICS JOURNAL

INDIANA UNIV MATH JOURNAL

Año: 2013 vol. 62 p. 890 - 910

0022-2518

The Calderon operator S is the sum of the the Hardyaveraging operator and its adjoint. The weights w for which Sis bounded on Lp(w) are the Calderon weights of the class Cp.We give a new characterization of the weights in Cp by a singlecondition which allows us to see that Cp is the class of Muckenhouptweights associated to a maximal operator dened through a basisin (0;1). The same condition characterizes the weighted weaktypeinequalities for 1 < p < 1, but that the weights for thestrong type and the weak type dier for p = 1. We also provethat the weights in Cp do not behave like the usual Ap weightswith respect to some properties and, in particular, we answer anopen question on extrapolation for Muckenhoupt bases without theopenness property.