Framework. Let $(X,\mu)$, and $(Y,\nu)$ be probability spaces on compact Hausdorff sets. Let $T:X\longrightarrow Y$ be a measurable function. Assume $\fbox{$T^{\ast}\mu\ll\nu$}$, where $T^{\ast}\mu$ is the pull-back measure on $Y$.${}^{\color{red}{\dagger}}$ Set
$$h:=\dfrac{\partial T^{\ast}\mu}{\partial\nu}\in L^{1}(Y,\nu).$$
W.l.o.g. $0\leq h<\|h\|_{\infty,\text{ess}}\leq\infty$ everywhere.
Question. What conditions on $T$ guarantee that the Radon-Nikoym derivative $h$ is (essentially) bounded?
Obviously a trivial sufficient condition would be $(X,\mu)=(Y,\nu)$ and $T=\mathrm{id}$. But this is uninteresting. Are there any results based on conditions in the direction of ‘$T$ $\mu$-a.e. Lipshitz-continuous’ or ‘$T$ diffeomorphism’, etc.? Note: the obvious condition $\exists{C>0:~}\forall{U\subseteq Y,~\text{open}:~}(T^{\ast}\mu)(U)\leq C\nu(U)$ is not something what I'm looking for, as this is too measure-dependent.
If the problem is too abstract, then feel free to $\fbox{just assume}$ / focus on:
- $X=[0,1]^{m}$, $Y=[0,1]^{n}$ for some $m,n\in\mathbb{N}^{+}$;
- $\mu,\nu$ are Lebesgue measures; and
- $T$ is at least continuous.
${}^{\color{red}{\dagger}}$ Defined as usual by $\int f~\mathrm{d}T^{\ast}\mu=\int f\circ T~\mathrm{d}\mu$ for all $f\in C(X)$ or equivalently $(T^{\ast}\mu)(A)=\mu(T^{-1}(A))$ for all $A\subseteq Y$ measurable.