Symmetric decreasing rearrangement

Given a measurable set, ${\displaystyle A}$, in Rⁿ , 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

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

where ${\displaystyle \omega _{n}}$ is the volume of the unit ball and where ${\displaystyle |A|}$ is the volume of ${\displaystyle A}$.

The rearrangement of a non-negative, measurable real-valued function ${\displaystyle f}$ whose level sets ${\displaystyle f^{-1}(y)}$ (${\displaystyle y\in \mathbb {R} _{\geq 0}}$) 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 ${\displaystyle f^{*}(x)}$ 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 ${\displaystyle g}$, the above definition is the unique definition that forces the identity ${\displaystyle \mathbb {I} _{A}^{*}=\mathbb {I} _{A^{*}}}$ to hold.

Function and its symmetric decreasing rearrangement preserve the measure of level sets.

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

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

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

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

The Hardy–Littlewood inequality holds, i.e.

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

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, i.e.

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

and

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

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.

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

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

We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement of f as the function ${\displaystyle f^{*}:[0,\infty )\to [0,\infty ]}$ and the rule

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

Note that this version of the decreasing rearrangement is not symmetric, as it is only defined on the nonnegative real numbers. However, it inherits many of the same properties listed above as the symmetric version. Namely:

f and f* are equimeasurable, i.e. they have the same distribution function.

The Hardy-Littlewood inequality holds, i.e. ${\displaystyle \int _{E}|fg|\;d\mu \leq \int _{0}^{\infty }f^{*}(t)g^{*}(t)\;dt.}$

${\displaystyle f\leq g}$ μ-a.e. implies ${\displaystyle f^{*}\leq g^{*}}$.

${\displaystyle (af)^{*}=|a|f^{*}}$ for all real numbers a.

${\displaystyle (f+g)^{*}(t_{1}+t_{2})\leq f^{*}(t_{1})+g^{*}(t_{2})}$ for all ${\displaystyle t_{1},t_{2}\in [0,\infty )}$.

${\displaystyle |f_{n}|\uparrow |f|}$ μ-a.e. implies ${\displaystyle f_{n}^{*}\uparrow f^{*}}$.

${\displaystyle (f^{p})^{*}=(f^{*})^{p}}$ for all positive real numbers p.

${\displaystyle \|f\|_{L_{p}(E)}=\|f^{*}\|_{L_{p}[0,\infty )}}$ for all positive real numbers 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 μ-a.e.

Note that the definitions of all the terminology in the above theorem (i.e., 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).

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