Approximation of two densities with a single transformation

Question

Let $p_1$ and $p_2$ be two probability densities and $X_i\sim N(\mu_i,\Sigma_i)$. Write $w(X)\sim p$ if the law of the random variable $w(X)$ has a density equal to $p$. For general densities $p_i$, is it possible to find $w$ and $(\mu_i,\Sigma_i)$ such that $w(X_i)\sim p_i$?

Note that if $w$ is allowed to depend on $i$ and $X_i$ is fixed to be standard normal (i.e. independent of $i$), this is trivially true. The interesting part is holding $w$ fixed but allowing the mean and variance of $X_i$ to vary. In fact, it is also true if the $X_i$ are uniformly distributed over disjoint intervals.

Bonus: If the answer is no, are there conditions on $p_i$ that are sufficient to ensure the answer is "yes"?

Answer 1

Not sure if I misunderstood but if you consider $w$ to be differentiable bijection, define $Y_i := w(X_i)$, denote the respective PDFs as $f_{X_{i}}$ and $f_{Y_{i}}\equiv p_i$, then the pushforward relations give $$\dfrac{f_{X_{1}}\circ w^{-1}}{f_{X_{2}}\circ w^{-1}} = \dfrac{p_1}{p_2},$$ where $\circ$ means function composition. This seems to be a necessary condition for $w$.