Muckenhoupt-type conditions on weighted Morrey spaces

We define a Muckenhoup-type condition on weighted Morrey spaces using the K\"othe dual of the space. We show that the condition is necessary and sufficient for the boundedness of the maximal operator defined with balls centered at the origin on weighted Morrey spaces. A modified condition characterizes the weighted inequalities for the Calder\'on operator. We also show that the Muckenhoup-type condition is necessary and sufficient for the boundedness on weighted local Morrey spaces of the usual Hardy-Littlewood maximal operator, simplifying the previous characterization of Nakamura-Sawano-Tanaka. For the same operator, in the case of global Morrey spaces the condition is necessary and for the sufficiency we add a local $A_p$ condition. We can extrapolate from Lebesgue $A_p$-weighted inequalities to weighted global and local Morrey spaces in a very general setting, with applications to many operators.