ABSTRACT
One of the important problems in the theory of function spaces is the boundedness property of operators. This is important because it yields many applications in various fields of mathematics such as partial differential equations and potential theory. This chapter discusses the Hardy–Littlewood maximal operator (including the sharp maximal operator) and fractional integral operators. Basically, the boundedness property of operators can be proved by the local estimate initiated by Burenkov and Guliyev. However, to make the proof self-contained we consider a direct proof which still recasts the flavor of the local estimates. The chapter presents a typical argument about the proof of the boundedness of operators in Morrey spaces. It ends with the boundedness of the Hardy–Littlewood maximal operator in predual spaces.
