Consider the space $\mathcal{P}(\mathbb{R})$ of Borel probability measures on $\mathbb{R}$. The $p$-order Cramer distance $l_p$ ($p \geq 1$) is defined by
$$ l_p(\mu, \nu) = \left(\int_{-\infty}^{\infty} |F_{\mu}(x) - F_{\nu}(x)|^p dx\right)^{1/p} $$ for all $\mu, \nu \in \mathcal{P}(\mathbb{R})$ where $F_{\mu}$ and $F_{\nu}$ denote the (cummulative) distribution functions of $\mu$ and $\nu$.
The $p$-order Wasserstein distance is defined by $$ W_p(\mu, \nu) = \left( \inf_{\pi \in \Pi(\mu,\nu)} \int_{\mathbb{R}^2} |x-y|^p d\pi(x,y) \right)^{1/p}, $$ where $\Pi(\mu,\nu)$ denotes the set of all Borel probability measures on $\mathbb{R}^2$ with marginals $\mu$ and $\nu$.
It is known that $W_1 = l_1$ (via their dual forms). But for $p > 1$, are these two metrics $W_p$ and $l_p$ compatible or at least could we bound one by the other in some way?
I came up with the same question! Yet, I haven't found anything else than this paper (https://arxiv.org/pdf/1705.10743.pdf) that says without references in section 4.1