Quantitative bound on Wasserstein distances by $L^p$ distances?

Question

Given two smooth probability densities $f$ and $g$ on $\mathbb{R}$ (or $\mathbb{R}_+$) with finite $p$-th moments. I am wondering if anyone is aware of some explicit upper bound on $W_p(f,g)$ in terms of $\|f-g\|_{L^p}$ (especially for $p=1$ and $p=2$) ? As one typically view Wasserstein distances as "weak" metrics and $L^p$ metrics as "strong" metrics. It is very natural to search for some explicit bound on $W_p(f,g)$ in terms of $\|f-g\|_{L^p}$. However, I did not find any useful references in this regard. Any help or pointers to literatures are greatly appreciated!

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Remark: It is well-known that $W_1(f,g)$ has a nice formulation in terms of cumulative distribution functions (while it seems that $W_2$ has no such nice representation). Indeed, let $F$ and $G$ represent the cumulative distribution functions corresponding to the densities $f$ and $g$, respectively. Then we have $$W_1(f,g) = \int |F(x)-G(x)| \mathrm{d}x$$ So in some sense, $W_1(f,g)$ looks like some $L^1$ norm (except that we are using the cdf's instead the pdf's). I am very curious as to why $$\int |F(x)-G(x)| \mathrm{d}x$$ can be somehow controlled by $$\int |f(x)-g(x)| \mathrm{d}x,$$ and if possible, I would like to see an explicit bound.

Answer 1

Indeed, there is a comparison, see Proposition 7.10 in Villani's "Topics in Optimal Transportation", which implies that if for example $f$ and $g$ are two probability measures in $L^1(\Bbb R^d)$ (that is absolutely continuous with respect to the Lebesgue measure), then for any $p\geq 1$, $$ W_p(f,g) \leq \max(1,2^{1-1/p}) \,\||x|^p(f-g)\|_{L^1(\Bbb R^d)}^{1/p} $$ If $f$ and $g$ are supported on a bounded domain, this implies that $W_p(f,g) \leq \|f-g\|_{L^p(\Bbb R^d)}$, but there is no special reason to take the $L^p$ norm here as far as I know?

In general one can think of the $W_p$ distances as close to negative Sobolev distances. For example, by the Kantorovich-Rubinstein formula, $W_1(f,g)$ is indeed nothing else but the $W^{-1,1}(\Bbb R^d)$ distance (that is the dual of $W^{1,\infty}(\Bbb R^d) = \mathrm{Lip}(\Bbb R^d)$). Then there are close links between $W_2$ and $\dot{H}^{-1}$ (see e.g. https://arxiv.org/pdf/1104.4631.pdf) and between $W_p$ and $B^{-1}_{p,q}$ (see https://q-berthet.github.io/papers/WeeBer19.pdf), where $B^s_{p,q}$ are very close to the spaces $W^{-1,p}$. However, these links usually need some uniform bound by above or below. Then by Sobolev embeddings, one can control negative Sobolev norms $W^{-1,p}$ by $L^r$ norms (for a smaller exponent $r = \frac{dp}{p+d}$), indicating that they have a weaker control locally but a stronger control at infinity than $L^p$ norms.