Symmetric decreasing rearrangement explained

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 \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$$A^*\; =\; \backslash left\backslash ,$$where

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

Definition for functions

The rearrangement of a non-negative, measurable real-valued function

f whose level sets f ^{-1} (y) (for y \in \R _{\geq} ) have finite measure is$$f^*(x)\; =\; \backslash int\_0^\backslash infty\; \backslash mathbb\_(x)\; \backslash ,\; dt,$$where 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 symmetricrearrangement of \{y: f(y) > t\} is equal to x. We have the following motivation for this definition. Because the identity$$g(x)\; =\; \backslash int\_0^\backslash infty\; \backslash mathbb\_(x)\; \backslash ,\; dt,$$holds for any non-negative function g, the above definition is the unique definition that forces the identity I ^{*} _{A} = 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, that is,$$|\backslash |\; =\; |\backslash |.$$

If

f is a function in L ^{p,} then$$\backslash |f\backslash |\_\; =\; \backslash |f^*\backslash |\_.$$

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

1 \leq p < inf ty and if f \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,$$f\; \backslash leq\; g\; \backslash text\; f^*\; \backslash leq\; g^*$$and$$\backslash |f\; -\; g\backslash |\_\; \backslash geq\; \backslash |f^*\; -\; g^*\backslash |\_.$$

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.

Nonsymmetric decreasing rearrangement

We can also define

f ^{*} as a function on the nonnegative real numbers rather than on all of \R ^{n.} ^[2] Let (E, \mu) be a σ-finite measure space, and let f : E\to[ -inf ty, inf ty] be a measurable function that takes only finite (that is, real) values μ-a.e. (where "\mu -a.e." means except possibly on a set of \mu -measure zero). We define the distribution function \mu _{f} : [ 0 , inf ty]\to[ 0 , inf ty] by the rule$$\backslash mu\_f(s)=\backslash mu\backslash .$$We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) of f as the function f ^{*} : [ 0 , inf ty)\to[ 0 , inf ty] by the rule$$f^*(t)=\backslash inf\backslash .$$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,

\int _{E|fg| d\mu\leq\int} ^{inf ty} ^{0} f ^{*(t)g} ^{*(t) dt.} \vert f\vert\leq\vert g\vert \mu -a.e. implies f ^{*\leq} g ^{*.} (af) ^{*=|a|f} ^{*} for all real numbers a. (f + g) ^{*(t} _{1+ t} _{2)} \leq f ^{*(t} _{2 )} for all t _{1,t} _{2\in[0,infty).} |f _{n|\uparrow|f|} \mu -a.e. implies f ^{*\uparrow} _{n} f ^{*.} \left(\vert f\vert ^{p\right)} ^{*=(f} ^{*)} ^{p} for all positive real numbers p. \|f\| _{L _{p(E)}} = \|f ^{*\|} _{L _{p[0,infty)}} for all positive real numbers p. \|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

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

Notes and References

1. Book: Lieb. Elliott. Elliott H. Lieb. Loss. Michael. Michael Loss. Analysis. 2001. 2nd. American Mathematical Society. Graduate Studies in Mathematics. 14. 978-0821827833.
2. Book: Bennett. Colin. Sharpley. Robert. Interpolation of Operators. 1988. 978-0-120-88730-9.