Symmetric decreasing rearrangement

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

An equivalent definition is

where $\backslash omega\_n$ is the volume of the unit ball and where $|A|$ is the volume of $A.$

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

$$f^*(x)\; =\; \backslash int\_0^\backslash infty\; \backslash mathbb\{I\}\_\{\backslash \{y:\; f(y)>t\backslash \}^*\}(x)\; \backslash ,\; dt,$$

where $\backslash mathbb\{I\}\_A$ 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 $\backslash \{y:\; f(y)\; >\; t\backslash \}$ is equal to $x.$ We have the following motivation for this definition. Because the identity

$$g(x)\; =\; \backslash int\_0^\backslash infty\; \backslash mathbb\{I\}\_\{\backslash \{y:\; g(y)>t\backslash \}\}(x)\; \backslash ,\; dt,$$

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

File:Function and its symmetric decreasing rearrangement.svg

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

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

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

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

The Hardy–Littlewood inequality holds, that is,

$$\backslash int\; fg\; \backslash leq\; \backslash int\; f^*\; g^*.$$

Further, the Pólya–Szegő inequality holds. This says that if and if $f\; \backslash in\; W^\{1,p\}$ then

$$\backslash |\backslash nabla\; f^*\backslash |\_p\; \backslash leq\; \backslash |\backslash nabla\; f\backslash |\_p.$$

The symmetric decreasing rearrangement is order preserving and decreases $L^p$ distance, that is,

$$f\; \backslash leq\; g\; \backslash text\{\; implies\; \}\; f^*\; \backslash leq\; g^*$$

and

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

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.

We can also define $f^*$ as a function on the nonnegative real numbers rather than on all of $\backslash R^n.${{cite book|last1=Bennett|first1=Colin|last2=Sharpley|first2=Robert|title=Interpolation of Operators|year=1988|isbn=978-0-120-88730-9}} Let $(E,\; \backslash mu)$ be a σ-finite measure space, and let $f\; :\; E\backslash to[-\backslash infty,\backslash infty]$ be a measurable function that takes only finite (that is, real) values μ-a.e. (where "$\backslash mu$-a.e." means except possibly on a set of $\backslash mu$-measure zero). We define the distribution function $\backslash mu\_f\; :\; [0,\backslash infty]\backslash to[0,\backslash infty]$ by the rule

$$\backslash mu\_f(s)=\backslash mu\backslash \{x\backslash in\; E\; :\; \backslash vert\; f(x)\backslash vert\; >s\backslash \}.$$

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

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

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, that is, they have the same distribution function.
• The Hardy-Littlewood inequality holds, that is, $\backslash int\_E|fg|\backslash ;d\backslash mu\backslash leq\backslash int\_0^\backslash infty\; f^*(t)g^*(t)\backslash ;dt.$
• $\backslash vert\; f\backslash vert\backslash leq\backslash vert\; g\backslash vert$ $\backslash mu$-a.e. implies $f^*\backslash leq\; g^*.$
• $(af)^*=|a|f^*$ for all real numbers $a.$
• $(f+g)^*(t\_1+t\_2)\; \backslash leq\; f^*(t\_1)+g^*(t\_2)$ for all $t\_1,t\_2\backslash in[0,\backslash infty).$
• $|f\_n|\backslash uparrow|f|$ $\backslash mu$-a.e. implies $f\_n^*\backslash uparrow\; f^*.$
• $\backslash left(\backslash vert\; f\backslash vert^p\backslash right)^*=(f^*)^p$ for all positive real numbers $p.$
• $\backslash |f\backslash |\_\{L\_p(E)\}=\backslash |f^*\backslash |\_\{L\_p[0,\backslash infty)\}$ for all positive real numbers $p.$
• $\backslash |f\backslash |\_\{L\_\backslash infty(E)\}=f^*(0).$

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

• {{annotated link|Isoperimetric inequality}}
• {{annotated link|Layer cake representation}}
• {{annotated link|Rayleigh–Faber–Krahn inequality}}
• {{annotated link|Riesz rearrangement inequality}}
• {{annotated link|Sobolev space}}
• {{annotated link|Szegő inequality}}

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{{Measure theory}}

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Category:Measure theory

Category:Multivariable calculus

Category:Real analysis