Let $p \in [1, \infty)$. Let $\mathcal P_p(\mathbb R^d)$ be the space of all Borel probability measures on $\mathbb R^d$ with finite $p$-th moments. 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 $\int_{\mathbb R^d} |x|^p f (x) \, \mathrm d x < \infty$. So $f \in D_p$ if and only if $f$ is a density of some $\mu \in \mathcal P_p(\mathbb R^d)$.

Let $F:D_p \to \mathcal P_p(\mathbb R^d)$ that sends a density $f$ to its corresponding $\mu \in \mathcal P_p(\mathbb R^d)$. We endow $\mathcal P_p(\mathbb R^d)$ with the $L_p$-Wasserstein metric $W_p$. We endow $D_p$ with the norm $[\cdot]$ defined by $$ [f-g]_p := \int_{\mathbb R^d} |f(x)-g(x)| \cdot |x|^p \, \mathrm d x \quad \forall f,g \in D_p. $$

Is $F$ Lipschitz? Are there some special properties about $F$?

Thank you so much for your elaboration?