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, ${\displaystyle A,}$ in ${\displaystyle \mathbb {R} ^{n},}$ one defines the symmetric rearrangement of ${\displaystyle A,}$ called ${\displaystyle A^{*},}$ as the ball centered at the origin, whose volume (Lebesgue measure) is the same as that of the set ${\displaystyle A.}$

An equivalent definition is

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

${\displaystyle \omega _{n))$

${\displaystyle |A|}$

${\displaystyle A.}$

Definition for functions

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

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

${\displaystyle \mathbb {I} _{A))$

${\displaystyle A.}$

${\displaystyle f^{*}(x)}$

${\displaystyle t}$

${\displaystyle \{y:f(y)>t\))$

${\displaystyle x.}$

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

${\displaystyle g,}$

${\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*))}$

Properties

The function ${\displaystyle f^{*))$ is a symmetric and decreasing function whose level sets have the same measure as the level sets of ${\displaystyle f,}$ that is,

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

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

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

The Hardy–Littlewood inequality holds, that is,

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

Further, the Pólya–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 ${\displaystyle L^{p))$ distance, that is,

$${\displaystyle f\leq g{\text{ implies ))f^{*}\leq g^{*))$$

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

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.

Nonsymmetric decreasing rearrangement

We can also define ${\displaystyle f^{*))$ as a function on the nonnegative real numbers rather than on all of ${\displaystyle \mathbb {R} ^{n}.}$^[2] Let ${\displaystyle (E,\mu )}$ be a σ-finite measure space, and let ${\displaystyle f:E\to [-\infty ,\infty ]}$ be a measurable function that takes only finite (that is, real) values μ-a.e. (where "${\displaystyle \mu }$-a.e." means except possibly on a set of ${\displaystyle \mu }$-measure zero). We define the distribution function ${\displaystyle \mu _{f}:[0,\infty ]\to [0,\infty ]}$ by the rule

$${\displaystyle \mu _{f}(s)=\mu \{x\in E:\vert f(x)\vert >s\}.}$$

${\displaystyle f}$

${\displaystyle f^{*}:[0,\infty )\to [0,\infty ]}$

$${\displaystyle f^{*}(t)=\inf\{s\in [0,\infty ]:\mu _{f}(s)\leq t\}.}$$

• ${\displaystyle f}$ and ${\displaystyle f^{*))$ are equimeasurable, that is, they have the same distribution function.
• The Hardy-Littlewood inequality holds, that is, ${\displaystyle \int _{E}|fg|\;d\mu \leq \int _{0}^{\infty }f^{*}(t)g^{*}(t)\;dt.}$
• ${\displaystyle \vert f\vert \leq \vert g\vert }$ ${\displaystyle \mu }$-a.e. implies ${\displaystyle f^{*}\leq g^{*}.}$
• ${\displaystyle (af)^{*}=|a|f^{*))$ for all real numbers ${\displaystyle a.}$
• ${\displaystyle (f+g)^{*}(t_{1}+t_{2})\leq f^{*}(t_{1})+g^{*}(t_{2})}$ for all ${\displaystyle t_{1},t_{2}\in [0,\infty ).}$
• ${\displaystyle |f_{n}|\uparrow |f|}$ ${\displaystyle \mu }$-a.e. implies ${\displaystyle f_{n}^{*}\uparrow f^{*}.}$
• ${\displaystyle \left(\vert f\vert ^{p}\right)^{*}=(f^{*})^{p))$ for all positive real numbers ${\displaystyle p.}$
• ${\displaystyle \|f\|_{L_{p}(E)}=\|f^{*}\|_{L_{p}[0,\infty )))$ for all positive real numbers ${\displaystyle p.}$
• ${\displaystyle \|f\|_{L_{\infty }(E)}=f^{*}(0).}$

The (nonsymmetric) decreasing rearrangement function arises often in the theory of rearrangement-invariant Banach function spaces. Especially important is the following:

Luxemburg Representation Theorem. Let ${\displaystyle \rho }$ be a rearrangement-invariant Banach function norm over a resonant measure space ${\displaystyle (E,\mu ).}$ Then there exists a (possibly not unique) rearrangement-invariant function norm ${\displaystyle {\overline {\rho ))}$ on ${\displaystyle [0,\infty )}$ such that ${\displaystyle \rho (f)={\overline {\rho ))(f^{*})}$ for all nonnegative measurable functions ${\displaystyle f:E\to [0,\infty ]}$ which are finite-valued ${\displaystyle \mu }$-a.e.

Note that the definitions of all the terminology in the above theorem (that is, Banach function norms, rearrangement-invariant Banach function spaces, and resonant measure spaces) can be found in sections 1 and 2 of Bennett and Sharpley's book (cf. the references below).