Symmetric decreasing rearrangement

In mathematics, the symmetric decreasing rearrangement of a function is a function which is symmetric and decreasing, and whose level sets are of the same size as those of the original function.^[1]

Definition for sets

Given a measurable set, $A$, in Rⁿ one can obtain the symmetric rearrangement of $A$, called $A^{*}$, by

${\displaystyle A^{*}=\{x\in \mathbf {R} ^{n}:\,\omega _{n}\cdot |x|^{n}<|A|\},}$

where $\omega _{n}$ is the volume of the unit ball and where $|A|$ is the volume of $A$. Notice that this is just the ball centered at the origin whose volume is the same as that of the set $A$.

Definition for functions

The rearrangement of a non-negative, measurable real-valued function $f$ whose level sets $f^{-1}(y)$ (${\displaystyle y\in \mathbb {R} _{\geq 0}}$) have finite measure is

${\displaystyle f^{*}(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:f(y)>t\}^{*}}(x)\,dt,}$

where ${\displaystyle \mathbb {I} _{A}}$ denotes the indicator function of the set A. In words, the value of $f^{*}(x)$ gives the height t for which the radius of the symmetric rearrangement of ${\displaystyle \{y:f(y)>t\}}$ is equal to x. We have the following motivation for this definition. Because the identity

${\displaystyle g(x)=\int _{0}^{\infty }\mathbb {I} _{\{y:g(y)>t\}}(x)\,dt,}$

holds for any non-negative function $g$, the above definition is the unique definition that forces the identity ${\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*}}}$ to hold.

Properties

The function $f^{*}$ is a symmetric and decreasing function whose level sets have the same measure as the level sets of $f$, i.e.

${\displaystyle |\{x:f^{*}(x)>t\}|=|\{x:f(x)>t\}|.}$

If $f$ is a function in $L^{p}$, then

${\displaystyle \|f\|_{L^{p}}=\|f^{*}\|_{L^{p}}.}$

The Hardy–Littlewood inequality holds, i.e.

${\displaystyle \int fg\leq \int f^{*}g^{*}.}$

Further, the Szegő inequality holds. This says that if ${\displaystyle 1\leq p<\infty }$ and if ${\displaystyle f\in W^{1,p}}$ then

${\displaystyle \|\nabla f^{*}\|_{p}\leq \|\nabla f\|_{p}.}$

The symmetric decreasing rearrangement is order preserving and decreases $L^{p}$ distance, i.e.

${\displaystyle f\leq g\Rightarrow f^{*}\leq g^{*}}$

and

${\displaystyle \|f-g\|_{L^{p}}\geq \|f^{*}-g^{*}\|_{L^{p}}.}$

Applications

The Pólya–Szegő inequality yields, in the limit case, with ${\displaystyle p=1}$, the isoperimetric inequality. Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality.

See also

• Isoperimetric inequality
• Layer cake representation
• Rayleigh–Faber–Krahn inequality
• Riesz rearrangement inequality
• Sobolev space
• Szegő inequality

References

1. Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.