Let $X=(X,d\_X)$ and $Y=(X,d\_Y)$ be metric spaces and $\varphi: X\rightarrow Y$ be an $L$-Lipschitz map, with $0 \le L < \infty$. Suppose $\mu$ is a probability measure on $X$ which satisfies *Talagrand transportation-cost inequality*, namely



> 
> There exists a constant $c\_\mu > 0$ such that
>  $$
> W(\nu,\mu) \le \sqrt{2c\_\mu H(\nu\|\mu)},
> $$
>  for every other probability measure $\nu$ on $X$.
> 
> 
> 


Here, $H$ denotes relative entropy (KL divergence) and $W$ denotes the Wasserstein $2$-distance on the space of probability measures on $X$ (with finite 2nd moment).


Question
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Does the pushforward $\varphi\_\#\mu$ of $\mu$ under $\varphi$ satisfy such an inequality ?
My guess is that it does, with constant $c\_{\varphi\_\#\mu} \le L c\_\mu$.



Transportation-cost inequality for pushforward measure
The result you want (but of course with $c\_{\varphi\_\#\mu} \le L^2 c\_\mu$ rather than $c\_{\varphi\_\#\mu} \le L c\_\mu$) is Lemma 2.1 in the paper at <https://arxiv.org/pdf/math/0410172>


