ABSTRACT
This chapter collects the properties of the linear operators acting on Lebesgue spaces. The boundedness obtained will be used for analysis of operators acting on Morrey spaces. This chapter discusses Hardy–Littlewood maximal operators, including the related maximal operators, Fractional maximal operators, Singular integral operators and Fractional integral operators. It gives some examples of calculations and address some important problems concerning its boundedness. The chapter deals with the Fefferman–Stein vector-valued inequality and investigates the boundedness property of the Orlicz maximal operator. One of the fundamental tools in the theory of function spaces is the Hardy–Littlewood maximal operator. A basic idea of analysis is the decomposition of a function into a countable sum of elementary pieces of functions. The Fefferman–Stein vector-valued maximal inequality is elementary and it is used mainly in the study of Triebel–Lizorkin spaces. One of the crucial properties of Morrey spaces is that Morrey spaces can describe the boundedness property of fractional integral operators more precisely than Lebesgue spaces.
