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:

$||\mu -\nu ||_{H^{-1}(\Omega )}={\text{sup}}{\Big \{}\int \phi d(\mu -\nu ):||\nabla \phi ||_{L^{2}}\leq 1{\Big \}}$

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: $b^{-{\frac {1}{2}}}||\mu -\nu ||_{H^{-1}(\Omega )}\leq W_{2}(\mu ,\nu )\leq a^{-{\frac {1}{2}}}||\mu -\nu ||_{H^{-1}(\Omega )}$