120 1. Measure theory Exercise 1.6.12 (Rising sun inequality). Let f : R R be an absolutely integrable function, and let f : R R be the one-sided signed Hardy- Littlewood maximal function f ∗(x) := sup h0 1 h [x,x+h] f(t) dt. Establish the rising sun inequality λm({f ∗(x) λ}) x:f ∗(x)λ f(x) dx for all real λ (note here that we permit λ to be zero or negative), and show that this inequality implies Lemma 1.6.16. (Hint: First do the λ = 0 case, by invoking the rising sun lemma.) See [Ta2009, §2.9] for some further discussion of inequalities of this type, and applications to ergodic theory (and in particular, the maximal ergodic theorem). Exercise 1.6.13. Show that the left- and right-hand sides in Lemma 1.6.16 are in fact equal. (Hint: One may first wish to try this in the case when f has compact support, in which case one can apply the rising sun lemma to a sufficiently large interval containing the support of f.) 1.6.2. The Lebesgue differentiation theorem in higher dimensions. Now we extend the Lebesgue differentiation theorem to higher dimensions. Theorem 1.6.11 does not have an obvious high-dimensional analogue, but Theorem 1.6.12 does: Theorem 1.6.19 (Lebesgue differentiation theorem in general dimension). Let f : Rd C be an absolutely integrable function. Then for almost every x Rd, one has (1.26) lim r→0 1 m(B(x, r)) B(x,r) |f(y) f(x)| dy = 0 and lim r→0 1 m(B(x, r)) B(x,r) f(y) dy = f(x), where B(x, r) := {y Rd : |x y| r} is the open ball of radius r centered at x. From the triangle inequality we see that | 1 m(B(x, r)) B(x,r) f(y) dy f(x)| = | 1 m(B(x, r)) B(x,r) f(y) f(x) dy| 1 m(B(x, r)) B(x,r) |f(y) f(x)| dy,
Previous Page Next Page