The Hardy–Littlewood Maximal Operator

Abstract

In this chapter we turn to the study of harmonic analysis on the variable Lebesgue spaces. Our goal is to establish sufficient conditions for the Hardy–Littlewood maximal operator to be bounded on L p(.); in the next chapter we will show how this can be used to prove norm inequalities on L p(.) for the other classical operators of harmonic analysis. We begin with a brief review of the maximal operator on the classical Lebesgue spaces and introduce our principal tool, the Calderon–Zygmund decomposition.