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 (more…)