A -density argument is useful in enlarging some sets a little bit while keeping the measure roughly the same.

Given a closed set and , a point is said to be -dense with respect to if

and the set of -dense points is denoted as . being close we see that and is closed as well. Even if we can somewhat reverse this: the measures and are in fact comparable, when finite.

Lemma 1There exists a constant such that

*Proof:* One just has to write explicitly,

where is a ball centered in and the condition then can be reformulated as

for some ball with center . But

where is the Hardy-Littlewood maximal function, thus the measure of is dominated by , which is in turn dominated by .

Notice the only point where we use that the measure is Lebesgue’s is when using the maximal function properties. Thus the theorem is true in a more general setting, in particular for doubling measures on metric spaces (see).

The lemma has the consequence that non-tangential maximal functions don’t actually depend on the aperture of the cone where you are taking the supremum. Say is a function . Then the non-tangential maximal function of parameter is

where the supremum is taken in both and , and therefore determines the aperture of this cone. It’s obvious that if then pointwise

but actually we have some inverse control on :

Lemma 2If then

Actually the control is directly over upper level sets. A quick remark: if instead of we consider then the above inequality becomes an inequality, provided the functions belong to for that exponent.

*Proof:* In this case one has to take

for a fixed . Notice that if then by definition there exist s.t. and . Then, since takes a supremum on a ball of radius , we have that the entire ball is contained inside (all points near have as neighbour!). This ball is further contained inside for obvious geometric reasons, and thus

Intersecting with instead of , if is chosen such that , we see that our (which was taken in ) is contained in , and thus with a constant depending on and (i.e. on ) we have

Since , it suffices to integrate on both sides.

To summarize the argument above, we proved that all points of have a small density in the complement of .

A useful observation is that the constant is given by .