On distance in total variation between image measures

We are interested in the estimation of the distance in total variation $$\mathrm{\Delta }:=\mathrm{\parallel }{P}_{f(X)}-P_{g(X)}{\mathrm{\parallel }}_{\mathrm{v}ar}$$ between distributions of random variables $f(X)$ and $g(X)$ in terms of proximity of $f$ and $g\mathrm{.}$ We propose a simple general method of estimating $\mathrm{\Delta }$. For Gaussian and trigonometrical polynomials it gives an asymptotically optimal result (when the degree tends to $\mathrm{\infty }$).