Given a measurable subset , we denote its -dimensional Lebesgue measure by .
Let be a bounded measurable set. Let be a measurable function. For , the level set is defined as
The sets , , and so on are defined by analogy. Then the distribution function of is given by
This function is a monotonically decreasing function of and for we have while for , we have . Thus the range of is the interval .
Definition (Decreasing rearrangement). Let be bounded and let be a measurable function. Then the (unidimensional) decreasing rearrangement of , denoted by , is defined on by
Essentially, is just the inverse function of the distribution function of . The following properties of the decreasing rearrangement are immediate from its definition.
Proposition 1. Let where is bounded. Then is a nonincreasing and left-continuous function.
Proposition 2. The mapping is non-decreasing, i.e. if in the sense that for all , where and are real-valued functions on then .
We now see that is indeed a rearrangement of .
Proposition 3. The function and are equimeasurable (i.e. they have the same distribution function), i.e. for all
Immediately one has
Corollary 1. With the preceding notations, we have
It is worth recalling from the theory of -functions that
Corollary 2. If and if for then and
In literature, we have the following general and powerful result
Theorem. Let be measurable. Let be a non-negative Borel measurable function. Then
We will end this first entry by providing several rearrangement inequalities.
Theorem ( continuity). Let . The mapping is continuous from into .
We prove this theorem directly and just a making use of Corollary 2. Next theorem is a very well-known theorem called the Hardy-Littlewood theorem.
Theorem (Hardy-Littlewood). Let and where and . Then
The proof of the Hardy-Littlewood inequality relies on a density argument and the following useful identity
Lemma. Let with integrable over . Let with . Then
We now see the fact that the rearrangement mapping is non-expansive between the relevant spaces.
Theorem (non-expansive property). Let where . Then
The proof of this theorem relies on a technique called truncation technique.
Similarly, we have the following variant
Definition (Increasing rearrangement). Let be bounded and let be a measurable function. Then the (unidimensional) increasing rearrangement of , denoted by , is defined on by
See also: Symmetrization And Applications (Series in Analysis) by S. Kesavan