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 .

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