We define a Muckenhoup-type condition on weighted Morrey spaces using the Köthe 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 Calderón 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.

我们使用空间的Köthe对偶在加权Morrey空间上定义Muckenhoup类型条件。我们证明了该条件对于以加权Morrey空间上的原点为中心的球所定义的最大算子的有界性是必要的和充分的。修改后的条件描述了Calderón算子的加权不等式。我们还表明，对于通常的Hardy–Littlewood最大算子的加权局部Morrey空间上的有界性，Muckenhoup型条件是必要的，并且足以满足要求，从而简化了Nakamura–Sawano–Tanaka的先前特征。对于同一运算符，在全局Morrey空间的情况下，该条件是必要的，并且为充分起见，我们添加了局部\（A_p \）条件。我们可以从勒贝格\（A_p \）推断加权不等式在一个非常笼统的环境中加权了全局和局部Morrey空间，并应用于许多运营商。