Characterization of $\BMO^{-1}$

By | June 10, 2018

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

  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

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