# Crawling ink spots lemma and its application

By | June 13, 2018

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

1. for all , there is a ball such that and 2. 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 .

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