ABSTRACT
In this chapter, the authors collect the properties of the linear operators acting on Lebesgue spaces. The boundedness obtained will be used for analysis of operators acting on Morrey spaces. The chapter discusses the following operators: Hardy–Littlewood maximal operators, including the related maximal operators, fractional maximal operators, singular integral operators, and fractional integral operators. It presents fundamental results for the Hardy–Littlewood maximal operator. The chapter gives some examples of calculations and address some important problems concerning its boundedness. It deals with the Fefferman–Stein vector-valued inequality. One of the fundamental tools in the theory of function spaces is the Hardy–Littlewood maximal operator. The Hardy–Littlewood maximal operator appears in many mathematical contexts. A basic idea of analysis is the decomposition of a function into a countable sum of elementary pieces of functions. The sum must be countable; otherwise the sum may fail to be measurable.
