I'm wondering if there is a way to bound $$ \left\| \int f dP - \int f dQ \right\| $$ by the $L_1$ distance between $P$ and $Q$: $$\int \left\| dP - dQ \right\| $$.
So far, I've only been able to show that $L_1(P,Q)$ is bounded from below by the squared Hellinger distance between $P$ and $Q$, but I don't see how to relate $L_1$ with integral probability metric.
Thanks!