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.

one defines the symmetric rearrangement of

as the ball centered at the origin, whose volume (Lebesgue measure) is the same as that of the set

An equivalent definition is

is the volume of the unit ball and where

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

whose level sets

) have finite measure is

denotes the indicator function of the set

In words, the value of

for which the radius of the symmetric rearrangement of

We have the following motivation for this definition.

holds for any non-negative function

the above definition is the unique definition that forces the identity

is a symmetric and decreasing function whose level sets have the same measure as the level sets of

The Hardy–Littlewood inequality holds, that is,

Further, the Pólya–Szegő inequality holds.

The symmetric decreasing rearrangement is order preserving and decreases

The Pólya–Szegő inequality yields, in the limit case, with

the isoperimetric inequality.

Also, one can use some relations with harmonic functions to prove the Rayleigh–Faber–Krahn inequality.

as a function on the nonnegative real numbers rather than on all of

be a σ-finite measure space, and let

be a measurable function that takes only finite (that is, real) values μ-a.e.

means except possibly on a set of

We define the distribution function

We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) of

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

Especially important is the following: 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.