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 1 There 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 2 If 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 .