Let $\Phi(x) = \pi^{-n/2}e^{-\abs{x}^2}$ be the Gaussian function, and $\Phi_t(x)=t^{-n}\Phi(x/t)$, where $x\in\mathbb{R}^n, t\in\mathbb{R}$. **Definition of the function space $BMO^{-1}$:** For any $v\in\mathcal{S}'(\mathbb{R}^n)$, we let $w=v*\Phi_{\sqrt{4t}}$, then we say $v$ belongs to the space $BMO^{-1}$ if the following norm is finite: \begin{equation} \norm{v}_{BMO^{-1}} = sup_{x,R}\big( \abs{B(x,R)}^{-1}\int_{B(x,R)} \int_0^{R^2} \abs{w}^2 dtdy \big)^{1/2} < \infty. \end{equation} === Translation and Scaling Invariant **Claim 9:** For any $\alpha>0, x_0\in\mathbb{R}^n$, we have $\norm{ v(x_0+x) }_{BMO^{-1}} = \norm{ v(x) }_{BMO^{-1}}$, and $\norm{ v(\alpha x) }_{BMO^{-1}} = \alpha^{-1}\norm{ v(x) }_{BMO^{-1}}$. This can be proved by direct computations which only involve variable change. **Claim 10:** Let the function space $Y$ be the one defined by (4) in [KT01], then for any $\alpha>0, y_0\in\mathbb{R}^n$, we have \begin{equation} \begin{split} \norm{ f(\sqrt{\alpha}y, \alpha t) }_Y &= \alpha^{-1} \norm{f(y,t)}_Y,~\\ \norm{ f(y+y_0, t)}_Y &=\norm{f(y,t)}_Y. \end{split} \end{equation} **Lemma 4:** Let $m\in\mathcal{C}^\infty(\mathbb{R}^n\backslash\{0\})$ be homogeneous of degree $0$. Then \begin{equation} \norm{ m(D_x) v }_{BMO^{-1}} \lesssim \norm{v}_{BMO^{-1}}, \end{equation} where $\mathcal{F}(m(D_x) v) = m(\xi)\hat v$. See [KT01], Lemma 4.1. === Question? We also have the point-wise estimate $\abs{w(x,t)} \lesssim t^{-1/2} \norm{v}_{BMO^{-1}}$. If you have a proof, please contact me (huangepn (at) gmail dot com). === References [KT01] H. Koch and D. Tataru, Well-posedness for the Navier-Stokes equations, Adv. Math. 157, 22-35 (2001)