is a locally integrable function in , define the maximal function as

here is an arbitrary ball.

Maximal operator is weakly (1,1) bounded. That is such that for ,

Let , is a compact set.

Then , where with . Apply the Vitali covering lemma

* Given a finite collection of balls in , then there exists a subcollection , disjoint from each other and *

We can conclude that . So

Since is arbitrary, we know is bounded by the right hand side.

Actually the decay of maximal function is a little faster than $\frac{1}{\lambda}$

There exists a constant such that

Define

Then . This means

So

Advertisements