Monthly Archives: June 2018

Crawling ink spots lemma and its application

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 .

Characterization of $\BMO^{-1}$

We fix some notations. For with , we write

We denote by which is a collection of all open balls in . Let be an open subset of . By , we denote the homogeneous Sobolev space defined as the completion of the complex-valued functions in the seminorm . The dual space is .

The space `bounded mean oscillation’ is introduced by John and Nirenberg [2]. A locally integrable function is in if

One research direction on PDEs is to study the global well-posedness(GWP) for small data and local well-posedness(LWP) for large data . Note that the natural scaling of the Navier-Stokes equation is

(1)

So it is natural to find a function space which is scaling invariant in this scaling. As an example, and are scaling invariant in this scaling. Kato [3] proved GWP for small initial data and LWP for initial data of the Navier-Stokes equation in . Similar result holds for . This result is proved by Cannone [1] and Planchon [6]. In [4], Koch and Tataru considered this problem, which is the largest spaces for the well-posedness. For , note that

is scaling-invariant with respect to \eqref{eq:NS-scaling}. Motivated by this, to gurantee the well-posedness, we require satisfying

This is related to the Calerson measure characterization of the space .

See Stein [7].

Motivated by this, we define the norm by

The following theorem is proved in [4,Theorem 1].

Theorem 1. Let be a tempered distribution. Then if and only if there exist with .

In this note, we give an another characterization of which was proved by Maz’ya and Verbitsky [5]. In that paper, they proved the following the Helmholtz decomposition:

Theorem 2. Let . Suppose that there exists a constant such that

(2)

for all cube in . Then we have

where

The condition \eqref{eq:BMO-inverse-characterization} is motivated from the form boundedness of second-order elliptic operators. See [5] for more details. Now we state the main theorem of this note.

Theorem 3. Let be a tempered distribution in , . Then if and only if there exists a constant such that \eqref{eq:BMO-inverse-characterization} holds for all cube in .

Proof. By Theorem 2, we have

where and . Set . So and hence by Theorem 1.

Conversely, suppose that , where . Then for every supported on a cube , we have

for some constant which does not depend on and . Hence by duality, we have

Here constants does not depend on . This completes the proof.

Remark. There is a Littlewood-Paley characterization of .

References

1. Marco Cannone, A generalization of a theorem by Kato on Navier-Stokes
equations, Rev. Mat. Iberoamericana 13 (1997), no. 3, 515–541. MR 1617394
2. F. John and L. Nirenberg, On functions of bounded mean oscillation,  Comm. Pure Appl. Math. 14 (1961), 415–426. MR 0131498
3. Tosio Kato, Strong L p -solutions of the Navier-Stokes equation in R m , with applications to weak solutions, Math. Z. 187 (1984), no. 4, 471–480. MR
760047
4. Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843
5. V. G. Maz 0 ya and I. E. Verbitsky, Form boundedness of the general second-order differential operator, Comm. Pure Appl. Math. 59 (2006), no. 9, 1286–1329. MR 2237288
6. F. Planchon, Global strong solutions in Sobolev or Lebesgue spaces to the
incompressible Navier-Stokes equations in R 3 , Ann. Inst. H. Poincar´ e Anal. Non Lin´eaire 13 (1996), no. 3, 319–336. MR 1395675
7. Elias M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, vol. 43, Princeton University Press, Princeton, NJ, 1993, With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III. MR 1232192