ABSTRACT
Morrey spaces can describe the local properties of functions over Rn. This applies to a general metric measure space. This chapter considers two typical cases. The first is where the Lebesgue measure is replaced by a general Radon measure on Rn. The second is where the underlying space differs from Euclidean space. The chapter is interested in cases where the norm bound is independent of the weights. Such an estimate can be obtained if the weighted case is considered. The chapter establishes the theory of Lebesgue spaces in Euclidean space with general Radon measures, while focusing on the theory of Lebesgue spaces in metric measure spaces. Using the covering lemma, the chapter investigates the boundedness property of these maximal operators. As an application of the boundedness property, it establishes a differentiation theorem.
