Density argument

A {\gamma}-density argument is useful in enlarging some sets a little bit while keeping the measure roughly the same.
Given a closed set {A\subset \mathbb{R}^n} and {0<\gamma<1}, a point {x\in\mathbb{R}^n} is said to be {\gamma}-dense with respect to {A} if

\displaystyle \inf_{r>0} \frac{|A\cap B(x,r)|}{|B(x,r)|}>\gamma,

and the set of {\gamma}-dense points is denoted as {A^\ast}. {A} being close we see that {A^\ast \subset A} and is closed as well. Even if {\left(A^\ast\right)^{c}\supset A^{c}} we can somewhat reverse this: the measures {|\left(A^\ast\right)^{c}|} and {|A^{c}|} are in fact comparable, when finite.

Lemma 1 There exists a constant {C=C_{n,\gamma}} such that

\displaystyle \left|\left(A^\ast\right)^c\right|\leq C_{n,\gamma} \left|A^c\right|.

Proof: One just has to write {\left(A^\ast\right)^c} explicitly,

\displaystyle \left(A^\ast\right)^c=\left\{x\,:\, \exists B, \frac{|A\cap B|}{|B|}\leq \gamma\right\},

where {B} is a ball centered in {x} and the condition then can be reformulated as

\displaystyle \frac{|A^c \cap B|}{|B|}>1-\gamma

for some ball with center {x}. But

\displaystyle \frac{|A^c \cap B|}{|B|}\leq M\left(\chi_{A^c}\right),

where {M} is the Hardy-Littlewood maximal function, thus the measure of {\left(A^\ast\right)^c} is dominated by {|\{M\left(\chi_{A^c}\right)>1-\gamma\}|}, which is in turn dominated by {\frac{C_n}{1-\gamma}\|\chi_{A^c}\|_{L^1}=\frac{C_n}{1-\gamma}|A^c|}. \Box

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 {F(x,t)} is a function {F\,:\, \mathbb{R}^n\times [0,+\infty[\rightarrow \mathbb{R}}. Then the non-tangential maximal function of parameter {a} is

\displaystyle F^\ast_{a}(x):=\sup_{|x-y|<at}{|F(y,t)|},

where the supremum is taken in both {t} and {y}, and {a} therefore determines the aperture of this cone. It’s obvious that if {a>b} then pointwise

\displaystyle F^\ast_{a}(x)\geq F^\ast_{b}(x),

but actually we have some inverse control on {F^\ast_{a}(x)}:

Lemma 2 If {a>b} then

\displaystyle \int_{\mathbb{R}^n}{F^\ast_{a}(x)}\,dx\lesssim_{a,b} \int_{\mathbb{R}^n}{F^\ast_{b}(x)}\,dx.

Actually the control is directly over upper level sets. A quick remark: if instead of {F} we consider {|F|^p} then the above inequality becomes an {L^p} inequality, provided the functions belong to {L^p} for that exponent.
Proof: In this case one has to take

\displaystyle A:=\{x\,:\, F^\ast_{b}(x)>\alpha\}^c

for a fixed {\alpha}. Notice that if {x\in \{ F^\ast_{a}>\alpha\}} then by definition there exist {\overline{y},\overline{t}} s.t. {|x-\overline{y}|<a\overline{t}} and {|F\left(\overline{y},\overline{t}\right)|>\alpha}. Then, since {F^\ast_b} takes a supremum on a ball of radius {bt}, we have that the entire ball {B\left(\overline{y},b\overline{t}\right)} is contained inside {A^c} (all points near {\overline{y}} have {\overline{y}} as neighbour!). This ball is further contained inside {B\left(x,(a+b)\overline{t}\right)} for obvious geometric reasons, and thus

\displaystyle \frac{|A^c\cap B\left(x,(a+b)\overline{t}\right)|}{|B\left(x,(a+b)\overline{t}\right)|}\geq \frac{|B\left(\overline{y},b\overline{t}\right)|}{|B\left(x,(a+b)\overline{t}\right)|}=\left(\frac{b}{a+b}\right)^n.

Intersecting with {A} instead of {A^c}, if {\gamma} is chosen such that {\gamma> 1-\left(\frac{b}{a+b}\right)^n}, we see that our {x} (which was taken in {\{ F^\ast_{a}>\alpha\}}) is contained in {\left(A^\ast\right)^c}, and thus with a constant depending on {n} and {\gamma} (i.e. on {a, b}) we have

\displaystyle |\{ F^\ast_{a}>\alpha\}| \lesssim_{n,a,b} |A^c|=|\{ F^\ast_{b}>\alpha\}|.

Since {\int{|\{|f|>\alpha\}|}\,d\alpha=\|f\|_{L^1}}, it suffices to integrate on both sides. \Box
To summarize the argument above, we proved that all points of \{ F^\ast_{a}>\alpha\} have a small density in the complement of \{ F^\ast_{b}>\alpha\}.
A useful observation is that the constant is given by {c_n \left(\frac{a+b}{b}\right)^n}.

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