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

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

Proposition

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: Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle b^{-\frac{1}{2}} || \mu - \nu ||_{H^{-1} \(Omega)} \leq W_2( \mu, \nu) \leq a^{-\frac{1}{2}}|| \mu - \nu ||_{H^{-1}\(Omega)} }