DefinitionWe define the Hardy-Littlewood Maximal function of to be

TheoremLet be a “weight” and associate a measure via . For measure ,

- , true also for .
- (Weak-type 1,1 bound)

Remark

- Assertion (1) is trivial and assertion (3) follows from (2) and Marcinkiewicz interpolation.
- If , then , and in this case we recover the result from a standard real analysis course.
- If , we say that obeys the
-conditionand as well as . The result is only true when (note that , and so ). A necessary and sufficient condition for is known and is called the condition:uniformly over all balls . Note when . It is know that if , then for some (cf Reverse Hölder’s inequality).

- Let be a metric measure space with a Borel measure . The proof will show that when , then
maps provided is

doubling, i.e., uniformly in and .

a.e convergence is equiv to boundedness of some sort of a maximal operator

Lemma (Vitali-type covering lemma)Given a finite collection of balls , there is a subcollection that is disjoint and

where then .

*Proof:* Run the following algorithm

- Take the largest (or equal) ball remaining and add that to .
- Discard (forever more) any ball that intersects our chosen one.
- If balls remain, go back to step 1, else stop.

By construction, the balls in are disjoint. Also, all of the balls that meet the chosen one in a particular iteration must be smaller than , and so will be contain and all of its neighboring balls.

*Proof:* (of Assertion 2) By inner regularity it suffices to control the measure of an arbitrary compact , the latter of which is an open set. Note that if then there is an so that

By compactness, we can cover by finitely many of these balls. We then apply the covering lemma to get a disjoint family with .

Now

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