Calderon weights as Muckenhoupt weights Javier DuoandikoetxeaFrancisco Martin-reyesS. Ombrosi 42B2540A3026D15Calder\\\'on operatorweighted inequalitiesmaximal operatorMuckenhoupt bases 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. Indiana University Mathematics Journal 2013 text pdf 10.1512/iumj.2013.62.4971 10.1512/iumj.2013.62.4971 en Indiana Univ. Math. J. 62 (2013) 891 - 910 state-of-the-art mathematics http://iumj.org/access/