IUMJ

Title: Calderon weights as Muckenhoupt weights

Authors: Javier Duoandikoetxea, Francisco J. Martin-Reyes and Sheldy Ombrosi

Issue: Volume 62 (2013), Issue 3, 891-910

Abstract:

The Calder\'on 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 Calder\'on weights of the class $\mathcal{C}_p$. We prove a characterization of the weights in $\mathcal{C}_p$ by a single condition which allows us to see that $\mathcal{C}_p$ is the class of Muckenhoupt weights associated with a maximal operator defined through a basis in $(0,\infty)$. The same condition characterizes the weighted weak-type inequalities for $1<p<\infty$, but that the weights for the strong type and the weak type differ for $p=1$. We also prove that the weights in $\mathcal{C}_p$ do not behave like the usual $A_p$ weights with respect to some properties and, in particular, we answer an open question on extrapolation for Muckenhoupt bases without the openness property.