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

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 $f^{-1}(y)$ ( $y\in \mathbb {R} _{\geq 0}$ ) have finite measure is

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

where $\mathbb {I} _{A}$ 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 $\{y:f(y)>t\}$ is equal to x. 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 $g$ , 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.

References

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

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