Let $n\in\mathbb N$, let $\lambda\in\mathbb R$, and let $V\in C^2(\mathbb R^n;\mathbb R)$ be such that the measure $\nu$ defined by $d\nu(x)=e^{-V(x)}\,d\mathcal L^n(x)$ is a Borel probability measure on $\mathbb R^n$ with finite second moment. Assume that
for every $x\in\mathbb R^n$ and every $\xi\in\mathbb R^n$.
paragraph
admin
Let $\mu\in\mathcal P_2(\mathbb R^n)$ satisfy $\mu=f\nu$ for a non-negative Borel density $f:\mathbb R^n\to[0,\infty)$. Suppose that the relative entropy
is finite, with the convention $0\log 0=0$. Suppose also that $\sqrt f\in W^{1,2}_{\mathrm{loc}}(\mathbb R^n)$ and that its weak gradient $\nabla\sqrt f:\mathbb R^n\to\mathbb R^n$ satisfies
Assume moreover the weighted Fisher-information recovery property for the triple $(V,\nu,\mu)$: there exist smooth strictly positive probability densities $f_k:\mathbb R^n\to(0,\infty)$ such that, with $\mu_k=f_k\nu$, the smooth HWI argument applies to each $\mu_k$ and