1.6. Differentiation theorems 125 summing in j and using the disjointness of the Bj we conclude that m( n i=1 Bi) 3d λ Rd |f(y)| dy. Since the B1,...,Bn cover K, we obtain Theorem 1.6.20 as desired. Exercise 1.6.19. Improve the constant 3d in the Hardy-Littlewood max- imal inequality to 2d. (Hint: Observe that with the construction used to prove the Vitali covering lemma, the centers of the balls Bi are contained in m j=1 2Bj and not just in m j=1 3Bj. To exploit this observation one may need to first create an epsilon of room, as the centers are not by themselves sufficient to cover the required set.) Remark 1.6.24. The optimal value of Cd is not known in general, although a fairly recent result of Melas [Me2003] gives the surprising conclusion that the optimal value of C1 is C1 = 11+ 61 12 = 1.56 . . .. It is known that Cd grows at most linearly in d, thanks to a result of Stein and Str¨ omberg [StSt1983], but it is not known if Cd is bounded in d or grows as d ∞. Exercise 1.6.20 (Dyadic maximal inequality). If f : Rd C is an abso- lutely integrable function, establish the dyadic Hardy-Littlewood maximal inequality m({x Rd : sup x∈Q 1 |Q| Q |f(y)| dy λ}) 1 λ R |f(t)| dt where the supremum ranges over all dyadic cubes Q that contain x. (Hint: The nesting property of dyadic cubes will be useful when it comes to the covering lemma stage of the argument, much as it was in Exercise 1.1.14.) Exercise 1.6.21 (Besicovich covering lemma in one dimension). Let I1,...,In be a finite family of open intervals in R (not necessarily disjoint). Show that there exist a subfamily I1,...,Im of intervals such that: (i) n i=1 In = m j=1 Im. (ii) Each point x R is contained in at most two of the Im. (Hint: First refine the family of intervals so that no interval Ii is contained in the union of the the other intervals. At that point, show that it is no longer possible for a point to be contained in three of the intervals.) There is a variant of this lemma that holds in higher dimensions, known as the Besicovitch covering lemma. Exercise 1.6.22. Let μ be a Borel measure (i.e., a countably additive mea- sure on the Borel σ-algebra) on R, such that 0 μ(I) for every interval I of positive length. Assume that μ is inner regular, in the sense that μ(E) = supK⊂E, compact μ(K) for every Borel measurable set E. (As
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