fa.functional analysis – Confusing definition of homogeneous Sobolev norm of order -1

Let $Omega subset mathbb{R}^{d}$ and $|.|$ is the standard euclidean $2$-norm. I came across a definition of $dot{H}^{-1}(Omega)$ which is a bit confusing. In (1) authors define the following semi-norm for $hin C^{1}(Omega)$:
begin{equation}
|h|_{dot{H}^{1}}:=left(int_{Omega} |nabla h(x)|_2^{2} dx right)^{1/2}
end{equation}

where $dx$ writes for the Lebesgue measure. They define for a signed measure $nu$ on $Omega$ the norm $|nu|_{dot{H}^{-1}(lambda)}$ by standard duality arguments:
begin{equation}
|nu|_{dot{H}^{-1}}=sup_{|h|_{dot{H}^{1}}leq 1} |int_{Omega} h d nu|= sup_{|h|_{dot{H}^{1}}leq 1} |langle h,nurangle|
end{equation}

where I noted $langle h,nurangle=int_{Omega} h d nu$ which defines a inner product.
Authors argue that this Homogeneous Sobolev norm is finite for measure having zero total mass.

I was a bit confused: is this definition a particular case of the more “standard” one using tempered distributions and Fourier transform (see Definition 1.31 in (2) e.g.) ? More precisely consider a tempered distribution $u$ over $Omega$ and:
begin{equation}
|u|_{dot{H}_*^{-1}}:= left(int_{Omega} |omega|^{-2}|hat{u}(omega)|^{2} d omegaright)^{1/2}
end{equation}

where $hat{u}$ writes for the Fourier transform. My idea is that if $nu$ is a signed measure zero total mass and with density wrt the Lesbegue measure, then it can be written as $nu= (f-g) dx$ where $f,g$ are positive functions with $int f dx=int g dx$. It can be seen as a the following tempered distribution:
begin{equation}
forall phi in S(Omega), langle nu,phirangle=int_{Omega} phi(x)d nu(x)=int_{Omega} phi(x)(f(x)-g(x))dx
end{equation}

where $S(Omega)$ is the Schwartz class. Moreover we would have something like $hat{nu}(omega)= (hat{f}(omega)-hat{g}(omega))$.

In a dirty way we would have also $|nabla h|_2^{2}=|omega|^{2} |hat{h}(omega)|^{2}$ so $|h|_{dot{H}^{1}}=| |omega| hat{h} |_{L_2}$. Moreover $|langle h,f-grangle|=|langle hat{h},hat{f}-hat{g}rangle|$ by Plancherel. And also $|langle hat{h},hat{f}-hat{g}rangle|=|langle |omega|^{1} hat{h},|omega|^{-1}(hat{f}-hat{g})rangle|leq | |omega|^{2} hat{h} |_{L_2} | |omega|^{-2}|hat{f}-hat{g}|^{2}|_{L_2}$. This would give:
begin{equation}
|nu|_{dot{H}^{-1}}= left(int_{Omega} |omega|^{-2}|hat{f}(omega)-hat{g}(omega)|^{2} d omegaright)^{1/2}=|nu|_{dot{H}_*^{-1}}
end{equation}

Does this reasoning make sense ? Does it requires additional assumptions on $h,f,g$ ?

(1) Comparison between W2 distance and H˙ −1 norm, and localisation of Wasserstein distance. Rémi Peyre. 2018.

(2) Fourier Analysis and Nonlinear Partial Differential Equations, Hajer BahouriJean-Yves CheminRaphaël Danchin. 2011