Gabriel–Popesco theorem

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.

Definition for sets

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

$A^{*}=\{x\in {\mathbf{R}}^n:{\omega }_n\cdot |x{|}^n<\left|A\right|\},$

where ${\omega }_n$ is the volume of the unit ball and where $\left|A\right|$ 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 function $f$ whose level sets have finite measure is

$f^{*}(x)=\underset{0}{\overset{\mathrm{\infty }}{\int }}{\mathbb{𝕀}}_{\{y:f(y)>t{\}}^{*}}(x)dt.$

We have the following motivation for this definition. Because the identity

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

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

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

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

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

The Hardy–Littlewood inequality holds, i.e.

$\int fg\le \int f^{*}{g}^{*}.$

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

$\Vert \mathrm{\nabla }{f}^{*}{\Vert }_p\le \Vert \mathrm{\nabla }f{\Vert }_p.$

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

$f\le g\Rightarrow f^{*}\le g^{*}$

and

$\Vert f-g{\Vert }_{L^p}\ge \Vert f^{*}-g^{*}{\Vert }_{L^p}.$

Applications

The Pólya–Szegő inequality yields, in the limit case, with $p=1$, 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

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.