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

$b= \nabla f + \Div F,$

where

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

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 $\BMO^{-1}$ .

References

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Herbert Koch and Daniel Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157 (2001), no. 1, 22–35. MR 1808843
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Characterization of $\BMO^{-1}$

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