Let $Y\subset\mathbb R^d$ and $X\subset\mathbb R$. Let $\mu\in\mathcal P(Y)$ be absolutely continuous with a density still denoted $\mu$ which is bounded with compact support (so $\mu(E)\leq\|\mu\|_{\infty}\mathcal L(E)$, where $\mathcal L$ is the Lebesgue measure). Let $\pi:Y\longrightarrow X$ be a Borel measurable function, which is continuous. Let $\nu\in\mathcal P(X)$ be the push-forward measure $\nu=\pi\#\mu$, which is absolutely continuous.
My question is: what are the conditions on $\pi$ (and on the measure $\mu$ if necessary) to have that the measure $\nu$ has a bounded (that is $L^{\infty}$) density? In other words, denoting by $\nu$ such a density, I want to write $$ \nu(E)\leq\|\nu\|_{\infty}\mathcal L(E). $$
A priori, I have that the density $\nu$ is only $L^1$.
Some help? Thank you