The crawling ink spots lemma is useful to exploit a level set argument. The argument is orginally due to Safonov and Krylov (1980).
Lemma 1. Let and be two open sets satisfying
- for all , there is a ball such that and
- for all such that
we have .
Then for some constant depending only on the dimension.
Proof. Let . Then by (i), we have a ball such that and
Since is open, choose a maximal ball such that , and . If , then by (i).
If , there is a ball such that . By maximality of , . Then
from the contrapositive of second condition. Now choose a decreasing sequence of balls converging to . Then by continuity of measure, we have
So we construct a cover of the set so that for all ,
Now we apply the Vitali covering lemma to get a countable subcollection of balls such that . Here denotes the cocentric ball with radious 5 times of the radius of ball . Since and , we have
This completes the proof of Lemma 1.
Application of crawling ink spots lemma
The crawling ink spots lemma helps us to get -estimates. The application of crawling ink spots lemma can be found at e.g. Dong and Kim (arXiv:1806.02635v1).
Fix . Suppose that . For , define
Let and . Suppose that there exists a constant such that the following hold: and , if
then we have
Then by Lemma 1, we see that
So we obtain the following estimates:
for some constant .