Let $P(x)$ and $Q(x)$ be two distributions of $\mathbf{x}$. Pinsker’s inequality says
$$D(P(x)\|Q(x)) \geq \frac{1}{2 \ln 2} \|P(x) - Q(x)\|_1^2,$$
where $\|\cdot\|_1$ is the $\mathcal{L}^1$ norm, and $D(P(x)\|Q(x))$ is the KL-divergence
$$D(P(x)\|Q(x)) = \int_{\mathcal{X}} P(x)\log_2 \frac{P(x)}{Q(x)} dx.$$
Given that $\int x^2 P(x) dx \neq \int x^2 Q(x) dx$ (Second moment constraint).
My question is when Pinsker’s inequality is tight. Thanks!
PS: My previous question is without the second moment contraint. In that case, $P = Q$ (a.e.) can make Pinsker's inequality tight.