Let
- $(E,\mathcal E,\lambda)$ be a measure space
- $\mu,\nu$ be measures on $\left(E^2,\lambda^{\otimes2}\right)$ with densities $f,g:E^2\to[0,\infty)$, respectively
- $\left\|\mu-\nu\right\|$ denote the total variation distance of $\mu$ and $\nu$
Note that \begin{equation} \begin{split} \left\|\mu-\nu\right\|&=\frac12\left\|f-g\right\|_{L^1\left(\lambda^{\otimes2}\right)}\\&=\frac12\sup_{\substack{h\::\:E^2\to\mathbb R\\h\text{ is }\mathcal E^{\otimes2}\text{-measurable}\\|h|\:\le\:1}}\left|\int\lambda({\rm d}x)\int\lambda({\rm d}y)\left(f(x,y)-g(x,y)\right)h(x,y)\right|. \end{split}\tag1 \end{equation}
Now let $h\in L^2(\lambda)$. Are we able to bounded the quantity $$\int\lambda({\rm d}x)\int\lambda({\rm d}y)|f(x,y)-g(x,y)||h(x)-h(y)|^2\tag2$$ in terms of $\left\|\mu-\nu\right\|$?