# decreasing rearrangements: why the asymmetry of measure-preserving maps?

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.

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I am a bit confused about your question. Could you make it more precise? – András Bátkai Aug 7 '13 at 15:35
@András Bátkai Are you familiar with the notion of a decreasing rearrangement of a function? – Mikhail Katz Aug 7 '13 at 15:37
I would say basic things I picked up at my university. I find the topic interesting but did not understand what you wanted to ask. What do you mean by "conceptual explanation"? There are many many reasons, but what is conceptual? – András Bátkai Aug 7 '13 at 15:43
To give an unrealistic example, if there were a natural cohomology group that gave an obstruction to finding $\psi$, that would be a conceptual explanation. Perhaps a more reasonable one would be to ask for a suitable norm such that precomposing by a measure-preserving map would always increase the norm. – Mikhail Katz Aug 7 '13 at 15:48