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 0<\delta<1 and E\subset F\subset B_{1} be two open sets satisfying

  1. for all x\in F, there is a ball B\subset B_{1} such that x\in B and

        \[ \left|E\cap B\right|\le\left(1-\delta\right)\left|B\right|. \]

  2. for all B\subset B_{1} such that

        \[ \left|E\cap B\right|>\left(1-\delta\right)\left|B\right|, \]

    we have B\subset F.

Then \left|E\right|\le\left(1-c\delta\right)\left|F\right| for some constant c depending only on the dimension.

Proof. Let x\in F. Then by (i), we have a ball B^{0}\subset B_{1} such that x\in B^{0} and

    \[ \left|E\cap B^{0}\right|\le\left(1-\delta\right)\left|B^{0}\right|. \]

Since F is open, choose a maximal ball B^{x} such that x\in B^{x}, B^{x}\subset B^{0} and B^{x}\subset F. If B^{x}=B^{0}, then \left|B^{x}\cap E\right|\le\left(1-\delta\right)\left|B^{x}\right| by (i).

If B^{x}\subsetneq B^{0}, there is a ball \tilde{B} such that B^{x}\subset\tilde{B}\subset B^{0}. By maximality of B^{x}, \tilde{B}\not\subset F. Then

    \[ \left|E\cap\tilde{B}\right|\le\left(1-\delta\right)\left|\tilde{B}\right| \]

from the contrapositive of second condition. Now choose a decreasing sequence of balls converging to B^{x}. Then by continuity of measure, we have

    \[ \left|E\cap B^{x}\right|\le\left(1-\delta\right)\left|B^{x}\right|. \]

So we construct a cover \left\{ B^{x}\right\} of the set F so that for all x\in F,

  • x\in B^{x};
  • B^{x}\subset F;
  • \left|B^{x}\cap E\right|\le\left(1-\delta\right)\left|B^{x}\right|.

Now we apply the Vitali covering lemma to get a countable subcollection of balls B_{j} such that F\subset\bigcup_{j=1}^{\infty}5B_{j}. Here 5B_{j} denotes the cocentric ball with radious 5 times of the radius of ball B_{j}. Since B_{j}\subset F and \left|E\cap B_{j}\right|\le\left(1-\delta\right)\left|B_{j}\right|, we have

    \begin{align*} \left|B_{j}\cap\left(F\setminus E\right)\right| & =\left|B_{j}\cap F\right|-\left|B_{j}\cap E\right|\\ & \ge\left|B_{j}\right|-\left(1-\delta\right)\left|B_{j}\right|\\ & =\delta\left|B_{j}\right|. \end{align*}


    \begin{align*} \left|F\setminus E\right| & \ge\sum_{j=1}^{\infty}\left|B_{j}\cap\left(F\setminus E\right)\right|\\ & \ge5^{-n}\delta\sum_{j=1}^{\infty}\left|5B_{j}\right|\\ & \ge5^{-n}\delta\left|F\right|. \end{align*}


    \begin{align*} \left|F\right| & =\left|F\setminus E\right|+\left|E\right|\\ & \ge5^{-n}\delta\left|F\right|+\left|E\right| \end{align*}

and so

    \[ \left|E\right|\le\left(1-5^{-n}\delta\right)\left|F\right|. \]

This completes the proof of Lemma 1.

Application of crawling ink spots lemma

The crawling ink spots lemma helps us to get \Leb{p}-estimates. The application of crawling ink spots lemma can be found at e.g. Dong and Kim (arXiv:1806.02635v1).

Fix 1<p<\infty. Suppose that D^2 u ,f \in \Leb{p}(\mathbb{R}^n). For s>0, define

    \[ \mathcal{A}(s) = \{ x \in \mathbb{R}^n : |D^2 u(x)|>s \} \]


    \[ \mathcal{B}(s) = \{ x \in \mathbb{R}^n : Mf(x)>s \}. \]

Let 0<R<\infty and 0<\gamma<1. Suppose that there exists a constant \kappa>1 such that the following hold: x_0 \in \mathbb{R}^n and s>0, if

    \[ |B_{R/4}(x_0) \cap \mathcal{A}(\kappa s)| \ge \gamma |B_{R/4}(x_0)|, \]

then we have

    \[ B_{R/4} (x_0) \subset \mathcal{B}(s). \]

Then by Lemma 1, we see that

    \[ |\mathcal{A}(ks)| \le (1-c\gamma)|\mathcal{B}(s)|. \]


    \[ \norm{f}{p}^p = \int_0^\infty p \lambda^{p-1} | \{ x \in \mathbb{R}^n : |f(x)|>\lambda \} | d\lambda, \]

we have

    \begin{align*} \norm{D^2 u}{p}^p &= \int_0^\infty p s^{p-1} |A(s) | ds \\ &=p\kappa^p \int_0^\infty s^{p-1} |A(\kappa s) | ds \\ &\leq C \int_0^\infty s^{p-1} |\mathcal{B}(s)| ds\\ &=C \norm{Mf}{p}^p \\ &\leq C \norm{f}{p}^p. \end{align*}

So we obtain the following estimates:

    \[ \norm{D^2 u}{p} \leq C \norm{f}{p} \]

for some constant C>0.

Characterization of $\BMO^{-1}$

We fix some notations. For A\subset \mathbb{R}^n with |A|<\infty, we write

    \[ f_A = \frac{1}{|A|} \int_A f dx. \]

We denote by \mathbb{B} which is a collection of all open balls in \mathbb{R}^n. Let \Omega be an open subset of \mathbb{R}^n. By \hSob{1}{2}(\Omega), we denote the homogeneous Sobolev space defined as the completion of the complex-valued C^\infty_0 functions in the seminorm \norm{\nabla u}{\Leb{2}(\Omega)}. The dual space is \hSob{-1}{2}(\Omega) = \hSob{1}{2}(\Omega)^*.

The space `bounded mean oscillation’ is introduced by John and Nirenberg [2]. A locally integrable function f:\mathbb{R}^n \rightarrow \mathbb{R} is in \BMO(\mathbb{R}^n) if

    \[ \sup_{B\in \mathbb{B}} \frac{1}{|B|} \int_B |f - f_B | dx<\infty. \]

One research direction on PDEs is to study the global well-posedness(GWP) for small data and local well-posedness(LWP) for large data u_0. Note that the natural scaling of the Navier-Stokes equation is

(1)   \begin{equation*} u_\lambda (x,t) =\lambda u(\lambda x, \lambda^2 t)\quad p_\lambda (x,t) = \lambda^2 p(\lambda x,\lambda^2 t),\quad\text{for } \lambda>0.\end{equation*}

So it is natural to find a function space which is scaling invariant in this scaling. As an example, \Leb{n} and \dot{B}_{p,q}^{n/p-1} (1<p<\infty) 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 \Leb{n}. Similar result holds for \dot{B}_{p,q}^{n/p-1} (1<p<\infty). 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 u \in \Leb{2}_{\loc}(\mathbb{R}^n\times [0,\infty)), note that

    \[ \sup_{x,R>0} \frac{1}{|B_R(x)|} \int_{B_R(x) \times [0,R^2]} |u|^2 dy dt \]

is scaling-invariant with respect to \eqref{eq:NS-scaling}. Motivated by this, to gurantee the well-posedness, we require e^{t\triangle}u_0 satisfying

    \[ \sup_{x,R>0} \frac{1}{|B_R(x)|} \int_{B_R(x) \times [0,R^2]} |e^{t\triangle} u_0|^2 dy dt \]

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

    \[ \norm{u}{\BMO} = \sup_{x,R>0} \left( \frac{1}{|B_R(x)|} \int_{B_R(x) \times [0,R^2]} |\nabla (e^{t\triangle }u_0)|^2 dy dt \right)^{1/2}. \]

See Stein [7].

Motivated by this, we define the \BMO^{-1} norm by

    \[ \norm{u}{\BMO^{-1}} = \sup_{x,R>0} \left( \frac{1}{|B_R(x)|} \int_{B_R(x) \times [0,R^2]} |e^{t\triangle }u_0|^2 dy dt \right)^{1/2}. \]

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

Theorem 1. Let f be a tempered distribution. Then f \in \BMO^{-1} if and only if there exist f^i \in \BMO with u = \sum_{i=1}^n \partial_i f^i.

In this note, we give an another characterization of \BMO^{-1} which was proved by Maz’ya and Verbitsky [5]. In that paper, they proved the following the Helmholtz decomposition:

Theorem 2. Let b\in \mathcal{D}'(\mathbb{R}^n). Suppose that there exists a constant C>0 such that

(2)   \begin{equation*} \norm{b}{\hSob{-1}{2}(Q)} \leq C |Q|^{\frac{1}{2}} \end{equation*}

for all cube Q in \mathbb{R}^n. Then we have

    \[ b= \nabla f + \Div F, \]


    \[ f=\triangle^{-1} \Div b \in \BMO,\quad F=\triangle^{-1} \curl b \in \BMO. \]

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 b be a tempered distribution in \mathbb{R}^n, n\geq 2. Then b \in \BMO^{-1} if and only if there exists a constant C>0 such that \eqref{eq:BMO-inverse-characterization} holds for all cube Q in \mathbb{R}^n.

Proof. By Theorem 2, we have

    \[ b= \nabla f +\Div F \]

where F \in \BMO^n and f\in \BMO. Set F_1 = F + \mathrm{diag} (\partial_1,\dots,\partial_n f) \in \BMO^{n\times n}. So b=\Div F_1 and hence b \in \BMO^{-1} by Theorem 1.

Conversely, suppose that b=\Div F, where F\in \BMO^{n\times n}. Then for every v\in C_0^\infty(Q) supported on a cube Q, we have

    \[ \norm{\nabla v}{\mathcal{H}^1} \leq C |Q|^{\frac{1}{2}} \]

for some constant C>0 which does not depend on Q and v. Hence by \mathcal{H}^1\BMO duality, we have

    \[ \left|\action{b,v}\right| = \left|\action{F,\nabla v}\right| \leq c \norm{F}{\BMO} \norm{\nabla v}{\mathcal{H}^1} \leq C |Q|^{\frac{1}{2}} \norm{\nabla v}{\Leb{2}(Q)}. \]

Here constants c,C does not depend on v. This completes the proof.

Remark. There is a Littlewood-Paley characterization of \BMO^{-1}.


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