Comparison between $W_2$ distance and $\dot{H}^{-1}$ norm, and localisation of Wasserstein distance

It is well known that the quadratic Wasserstein distance $W_2 (\mathord{\boldsymbol{\cdot}}, \mathord{\boldsymbol{\cdot}})$ is formally equivalent, for infinitesimally small perturbations, to some weighted $H^{-1}$ homogeneous Sobolev norm. In this article I show that this equivalence can be integrated to get non-asymptotic comparison results between these distances. Then I give an application of these results to prove that the $W_2$ distance exhibits some localisation phenomenon: if $μ$ and $ν$ are measures on $\mathbf{R}^n$ and $φ\colon \mathbf{R}^n \to \mathbf{R}_+$ is some bump function with compact support, then under mild hypotheses, you can bound above the Wasserstein distance between $φ\cdot μ$ and $φ\cdot ν$ by an explicit multiple of $W_2 (μ, ν)$.

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