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

Thus,

Hence,

and so

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

and

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

Since

we have

So we obtain the following estimates:

for some constant .