Symmetric decreasing rearrangement

Remarks

The function $f^*$ is nonnegative, radial, non-increasing in $\abs{x}$ and lower semicontinuous. For every $t > 0$, $\set{f^* > t} = \set{\abs{f} > t}^*$, so $f^*$ and $\abs{f}$ have the same distribution function, and $\norm{f^*}_p = \norm{f}_p$ for $1 \le p \le \infty$. The Pólya–Szegő inequality states that $\norm{\nabla f^*}_2 \le \norm{\nabla f}_2$ for $f \in H^1(\R^n)$. The Riesz rearrangement inequality states that $\iint f(x) g(x - y) h(y) \, dx \, dy \le \iint f^*(x) g^*(x - y) h^*(y) \, dx \, dy$ for nonnegative $f, g, h$.

Example: in $\R$, let $f = \mathbf{1}_{[1, 3]}$. For $0 < t < 1$ the set $\set{f > t}$ is $[1, 3]$, of length $2$, so its rearrangement is $(-1, 1)$; for $t \ge 1$ it is empty. Hence $f^* = \mathbf{1}_{(-1, 1)}$.

Non-example: the constant function $f = 1$ on $\R^n$ does not vanish at infinity, since $\set{f > 1/2}$ has infinite measure, so $f^*$ is not defined.

The one-variable decreasing rearrangement $f^\sharp$ on $[0, \infty)$ is related by $f^*(x) = f^\sharp(\omega_n \abs{x}^n)$, where $\omega_n$ is the volume of the unit ball, up to values on a null set.