Optimal Transport Wiki:Asymptotic equivalence of W 2 and H^-1

Asymptotic Equivalence of $W_{2}$ and $H^{-1}$

Equivalence

Definition of $H^{-1}$

The negative Sobolev norm $H^{-1}$ is defined:

Failed to parse (syntax error): {\displaystyle || \mu - \nu ||_{H^{-1} (\Omega)} = sup \big{ \int \phi d( \mu - \nu ) : || \nabla \phi ||_{L^2} \leq 1 \big} }

Theorem

Let $\mu ,\nu$ be absolutely continuous measures on a convex domain $\Omega$, with densities bounded from below and from above by constants $a,b$ with $0<a<b<+\infty$. Then: $b^{-{\frac {1}{2}}}||\mu -\nu ||_{H^{-1}(\Omega )}\leq W_{2}(\mu ,\nu )\leq a^{-{\frac {1}{2}}}||\mu -\nu ||_{H^{-1}(\Omega )}$