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

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

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 function $f$ whose level sets have finite measure is

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

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

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

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

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

The Hardy–Littlewood inequality holds, i.e.

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

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

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

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

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

and

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

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

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