 Symmetric decreasing rearrangement

The symmetric decreasing rearrangement $u^{\ast }$ of a measurable function is the radial, radially nonincreasing function whose superlevel sets are centered balls with the same measures as those of $|u|$. It preserves $L^p$ norms, and the Pólya-Szegő inequality gives $\Vert \nabla u^{\ast }{\Vert }_2\le \Vert \nabla u{\Vert }_2$.

Symmetric decreasing rearrangement is a mathematical concept used primarily in the field of analysis, particularly in the study of functions and measures. It is a technique that involves rearranging a sequence or a measurable function in such a way that the new arrangement is symmetric and non-increasing (i.e., it decreases or stays constant).