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. There is a standard technique which is a local/global strategy. Basically, the boundedness property of operators can be proved by the local estimate initiated by Burenkov and Guliyev. One of the prominent roles of Morrey spaces is that Morrey spaces can describe the local regularity and the global regularity more precisely than Lebesgue spaces. This aspect can be seen from the sharp maximal inequality. The chapter defines the sharp maximal operator and then formulates the sharp maximal inequality. Morrey used this observation to elliptic differential equations. Later some important formulas which can be expressed in terms of singular integral operators came about.
