247A, Notes 2: The Hardy–Littlewood maximal function and applications

— 1. The Hardy-Littlewood maximal inequality — We work in Euclidean space with Lebesgue measure; we write instead of for the Lebesgue measure of a set . For any and let denote the open ball of radius centred at . Thus for instance . For any , we use to denote the dilate of around its centre by . For any , we define the averaging operators on for any locally integrable by It is not hard to see that these averages are well-defined, and are even continuous functions, for locally integrable . One can view as an averaging operator From Schur’s test or Young’s inequality (or Minkowski’s inequality) we know that these are contractions on every , : Thus the averages are uniformly bounded in size as varies. The fundamental Hardy-Littlewood maximal inequality asserts, roughly speaking, that they are also uniformly bounded in shape: Proposition 1 (Hardy-Littlewood maximal inequality) We have the the strong-type inequality for all and any , and also the weak-type inequality for any . The…