Ryff proved in 1970 that the decreasing rearrangement $f^*$ of a, say, continuous function $f:[0,1]\to\mathbb{R}$ admits a measure preserving map $\phi$ such that $f=f^*\circ\phi$. In general it is impossible to find a measure-preserving map $\psi$ such that $f^*=f\circ\psi$; Ryff's counterexample was reproduced at Existence of rearrangements of functions in $L^p([0,1])$ when given a measure preserving map The counterexample is simple and clear, yet one would like to have a more conceptual explanation of the asymmetry between the existence of $\phi$ and the nonexistence of $\psi$.
Note that passage to the decreasing rearrangement decreases the $L^p$ norm of the derivative (whenever defined) of $f$ by the Szegő inequality. Can this be generalized to show that precomposing a function by a measure-preserving map always increases the $L^p$ norm of the derivative (whenever defined) of the function?
I noticed that in the introduction to chapter 3 in the book
Lieb, Elliott H.; Loss, Michael Analysis. Second edition. Graduate Studies in Mathematics, 14. American Mathematical Society, Providence, RI, 2001
the authors mention that passage to the monotone rearrangement decreases the "kinetic energy" of a function. This may be related to the Szegő inequality, but I am still not sure whether it is true more generally that precomposing by a measure-preserving transformation always increases these norms, or only in the case of the monotone rearrangement.
