Metrics on the space of distributions in terms of p.d.fs

Question

• If two probability distributions (on the same measure space) are s.t they have p.d.fs and the $L^1$ distance between the p.d.f.s is large, then is there a choice of a ``nice" metric $d_{\rm smooth}$ and a positive integer $p$ s.t the two distributions are far in the Wasserstein $p-$norm in $d_{\rm smooth}$?

  If we use the trivial metric $d(x,y) = {\bf 1}(x \neq y)$ then the ${\rm W}_{1,d} = 0.5 {\rm TV}$ and TV is the $L^1$ distance between the p.d.fs when they exist. But suppose that I want to use a smoother metric than this.

• Secondly if two probability distributions are such that their inverse CDFs are close (say in the sup norm) then are the two distributions close in some Wasserstein norm?

  Note that one can create examples where the two inverse CDFs are close in the sup norm but their TV distance is large hence their $1-$Wasserstein norm in the trivial metric is large.