Given a measurable subset , we denote its -dimensional Lebesgue measure by . We will denote by the open ball centered at the origin and having the same measure as , i.e. . The norm of vector will be denoted by . Finally, we will denote by the volume of the unit ball in . It is worth recalling that

where us the usual gamma function.

Definition(Schwarz symmetrization). Let be a bounded domain. Let be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function defined by.

Observe that if is the radius of , then

We obviously have the following properties of Schwarz symmetrization

- is radially symmetric and decreasing.
- are all equi-measurable.
- If is a Borel measurable function such that either or then
.

In particular, and have the same -norm and

.

when is integrable over .

- If is a non-decreasing function, then

. - The mapping is a non-expansive mapping from into for all .
- If is a measurable subset, then
.

Equality occurs iff

.

- The Hardy-Littlewood inequality holds, i.e.
for any and where and .

Similarly, we have the following variant.

Definition(Schwarz symmetrization). Let be a bounded domain. Let be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function defined by.

Now if we replace a bounded domain by the whole space , we have the following variant

Definition(Schwarz symmetrization for the whole space). Let be a measurable function which is vanishing at infinity in the sense that for any.

Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function defined by

.

Interestingly, preserves all properties decreasing rearrangement or increasing rearrangement. The most important property of this rearrangement is the Riesz inequality. The simplest form is the following

for any non-negative Borel measurable functions vanishing at infinity.

See also:

*Symmetrization And Applications*(Series in Analysis) by S. Kesavan- Symmetrization: The Decreasing Rearrangement
- Symmetrization (by Frank Morgan)

Hi, Ngo

I port to your attention this nice notes on rearrangement inequalities by Almut Burchard from Toronto.

http://www.math.toronto.edu/almut/rearrange.pdf

Comment by Fab — February 13, 2012 @ 19:18

Thanks Fab. A short but interesting note.

Comment by Ngô Quốc Anh — February 13, 2012 @ 22:34

Hi Ngo,

it will be nice to continue this post on Schwarz symmetrization with the classical Polya Szego inequality!

Comment by fab — March 7, 2012 @ 23:36