# 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