An standard way of characterizing elements of the function space BMO($\mathbb{R}^d$) is by means of Carleson measures. Namely we can consider the heat kernel $\Phi_t(x)=\pi^{-d/2}t^{-d}e^{-|x|^2/t}$ and with it define a "norm" (not really a norm) $$ \|f\|_{BMO} = \sup_{x\in\mathbb{R}^d, R>0} \left(\frac{2}{|B_R(x)|}\int_{B_R(x)}\int_0^{R^2} |f\ast\nabla\Phi_{\sqrt{4t}}|^2dtdy\right)^{1/2} \equiv \sup_{x\in\mathbb{R}^d, R>0} \left(\frac{2}{|B_R(x)|}\int_{B_R(x)}\int_0^{R^2} |\nabla\omega|^2dtdy\right)^{1/2} $$ with $\omega$ a solution to the heat equation with initial datum $f$, i.e. $\partial_t\omega -\Delta\omega = 0$ and $\omega(0,\cdot) = f$.
Based on this definition one can also consider a new space denoted by BMO$^{-1}$ with a similar norm (this time it is a true norm) with one derivative less for the solution to the heat equation, $$ \|f\|_{BMO^{-1}} = \sup_{x\in\mathbb{R}^d, R>0} \left(\frac{2}{|B_R(x)|}\int_{B_R(x)}\int_0^{R^2} |\omega|^2dtdy\right)^{1/2}. $$
We can also prove a correspondance between BMO and BMO$^{-1}$ because $f\in$ BMO$^{-1}$ if and only if there exists $g=(g_1,...,g_d)$ with $g_i\in$ BMO for all $i$ such that $f = \nabla\cdot g$.
This is all well known in Harmonic Analysis. My question is, I sort of understand why the notation for BMO$^{-1}$ because we consider "one derivative less", but is it by any chance also the dual (with respect to some topology) of the space BMO? The notation with the negative exponent is sometimes used to denote duality, but I don't see it straightforwardly here.