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.

Definition for sets

Given a measurable set, in one defines the symmetric rearrangement of called as the ball centered at the origin, whose volume is the same as that of the set
An equivalent definition is
where is the volume of the unit ball and where is the volume of

Definition for functions

The rearrangement of a non-negative, measurable real-valued function whose level sets have finite measure is
where denotes the indicator function of the set
In words, the value of gives the height for which the radius of the symmetric
rearrangement of is equal to We have the following motivation for this definition. Because the identity
holds for any non-negative function the above definition is the unique definition that forces the identity to hold.

Properties

The function is a symmetric and decreasing function whose level sets have the same measure as the level sets of that is,
If is a function in then
The Hardy–Littlewood inequality holds, that is,
Further, the Pólya–Szegő inequality holds. This says that if and if then
The symmetric decreasing rearrangement is order preserving and decreases distance, that is,
and

Applications

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.

Nonsymmetric decreasing rearrangement

We can also define as a function on the nonnegative real numbers rather than on all of Let be a σ-finite measure space, and let be a measurable function that takes only finite values μ-a.e.. We define the distribution function by the rule
We can now define the decreasing rearrangment of as the function by the rule
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:
  • and are equimeasurable, that is, they have the same distribution function.
  • The Hardy-Littlewood inequality holds, that is,
  • -a.e. implies
  • for all real numbers
  • for all
  • -a.e. implies
  • for all positive real numbers
  • for all positive real numbers
The 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 can be found in sections 1 and 2 of Bennett and Sharpley's book.