If a random vector $U \sim f_U(u)$ when does $W= f_U(U) $ have a bounded pdf? Suppose we have a random vector $U \in \mathbb{R}^n$ with a bounded pdf $f_U(u)$.  Now suppose we define a new random variable $W=f_U(U)$ with the pdf $f_W(w)$. 
What would be a condition on the random vector  $U$ or the pdf of $U$ such that the pdf of $f_W(w)$ is bounded? 
I tried going through the classical arguments ( for $n=1$), but I don't think the classical mapping theorem holds
\begin{align}
f_W(w)= f_U(f_U^{-1}(w)) \frac{d}{dw} f_U^{-1}(w)= w  \frac{d}{dw} f_U^{-1}(w)
\end{align}
since the pdf is not always invertable.  

It might be helpful to study this in the case when $f$ is monotonic, which implies it either has compact support or its support contains only one of $\pm\infty$. In that case $f_w(w)=w\frac{1}{f'_U(f_U^{-1}(w))}$, but it seems quite annoying to derive conditions, especially on the denominator that assure boundedness. For example if $f'_U(z)$ can be arbitrarily close to zero, this can cause problems,e.g. a half-normal or half-cauchy distribution. So in this situation if $f$ has compact support, then its enough for $f'_U(z)>\epsilon>0$ for all $z$. Otherwise if the support includes $+\infty$, then you would need $w/f'_U(f_U^{-1}(w))$ to stay bounded. 
