I have been looking for references on the following matter: let $f$ be the pdf of any real-value random variable ($f$ is not necessarily continuous wrt Lebesgue measure), and $g=g_{\mu,\sigma}$ be a Gaussian pdf. Writing $d_{\rm TV}$ for the total variation distance and $\ast$ for the convolution, is there anything that can be said between $d_{\rm TV}(f,g)$ and $d_{\rm TV}(g\ast f,g)$?
In particular, is there any result of the form $$ d_{\rm TV}(g\ast f,g) \leq \alpha\cdot d_{\rm TV}(f,g)^\beta $$ for some universal constants $\alpha,\beta >0$?
(if not, what would be a sufficient assumption on $f$? E.g., $f = a\mu + (1-a)\nu$ with $a > 0$ and $\mu$ a measure continuous wrt Lebesgue — would that be enough?)
If this is a straightforward exercise or a known and simple result (or, on the contrary, easily seen to be false), can someone point me towards either a proof of it, or a way/hint to prove it, or anything of that sort?
Edit: in case this could be easier (or hold while the above does not(?)), I am also interested in the same sort of result, but with the Earthmover's Distance (Wasserstein metric) instead of the more stringent total variation.
For the Wasserstein distance, perhaps Lemma 7.1.10 in the book ``Gradient Flows in Metric Spaces and the Space of Probability Measures'' by Ambrosio, Gigli, and Savaré is what you are looking for.