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.[1]

Definition for sets

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Given a measurable set,   in   one defines the symmetric rearrangement of   called   as the ball centered at the origin, whose volume (Lebesgue measure) 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

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The rearrangement of a non-negative, measurable real-valued function   whose level sets   (for  ) 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

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

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

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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

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We can also define   as a function on the nonnegative real numbers rather than on all of  [2] Let   be a σ-finite measure space, and let   be a measurable function that takes only finite (that is, real) values μ-a.e. (where " -a.e." means except possibly on a set of  -measure zero). We define the distribution function   by the rule   We can now define the decreasing rearrangment (or, sometimes, nonincreasing rearrangement) 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 (nonsymmetric) decreasing rearrangement function arises often in the theory of rearrangement-invariant Banach function spaces. Especially important is the following:

Luxemburg Representation Theorem. Let   be a rearrangement-invariant Banach function norm over a resonant measure space   Then there exists a (possibly not unique) rearrangement-invariant function norm   on   such that   for all nonnegative measurable functions   which are finite-valued  -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).

See also

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References

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  1. ^ Lieb, Elliott; Loss, Michael (2001). Analysis. Graduate Studies in Mathematics. Vol. 14 (2nd ed.). American Mathematical Society. ISBN 978-0821827833.
  2. ^ Bennett, Colin; Sharpley, Robert (1988). Interpolation of Operators. ISBN 978-0-120-88730-9.