Differentiation and the Maximal Operators

Abstract

We present an overview of certain maximal operators: the Hardy-Littlewood maximal op- erator and the dyadic maximal operator. We show that the two maximal operators are essentially the same. We study their weak type and strong type boundedness. We then use the weak type bounds of the Hardy-Littlewood maximal operator as a tool to differentiate integrals on dimensions higher than 1. Finally, we then study some construction of the dyadic systems in R