Fix $p \in [1, \infty)$. Let $(\mathcal P_p(\mathbb R^d), W_p)$ be the Wasserstein space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moment. Let $D_p$ be the collection of all Borel measurable functions $f:\mathbb R^d \to \mathbb R_{\ge 0}$ such that $\int_{\mathbb R^d} f (x) \, \mathrm d x = 1$ and $\int_{\mathbb R^d} f (x) |x|^p \, \mathrm d x < \infty$. So $f \in D_p$ if and only if $f$ is a density of some $\mu \in \mathcal P_p(\mathbb R^d)$.
Let $\Psi : D_p \to \mathcal P_p(\mathbb R^d)$ be the map that sends a density to its measure. For $\alpha \in (0, 1]$, we define $$ [f-g]_p^\alpha := \left ( \int_{\mathbb R^d} |f(x)-g(x)| \, |x|^p \, \mathrm d x \right ) ^\alpha, \quad \forall f,g \in D_p. $$
Are there constants $\alpha, C>0$ (possibly depending on $d, p$) such that $$ W_p (\Psi(f), \Psi(g)) \le C [f-g]_p^\alpha, \quad \forall f,g \in D_p. $$ ?
Thank you so much for your elaboration?
