Ngô Quốc Anh

May 2, 2010

Symmetrization: The Decreasing Rearrangement

Given a measurable subset E \subset \mathbb R^N, we denote its N-dimensional Lebesgue measure by |E|.

Let \Omega be a bounded measurable set. Let u :\Omega \to \mathbb R be a measurable function. For t \in \mathbb R, the level set \{u>t\} is defined as

\displaystyle \{u>t\}=\{x\in \Omega: u(x)>t\}.

The sets \{u<t\}, \{u \geqslant t\}, \{u=t\} and so on are defined by analogy. Then the distribution function of u is given by

\displaystyle \mu_u(t)=|\{u>t\}|.

This function is a monotonically decreasing function of t and for t \geq {\rm esssup}(u) we have \mu_u(t)=0 while for t\leqslant {\rm essinf}(u), we have \mu_u(t)=|\Omega|. Thus the range of \mu_u is the interval [0, |\Omega|].

Definition (Decreasing rearrangement). Let \Omega \subset \mathbb R^N be bounded and let u :\Omega \to \mathbb R be a measurable function. Then the (unidimensional) decreasing rearrangement of u, denoted by u^\sharp, is defined on [0, |\Omega|] by

\displaystyle {u^\sharp }(s) = \begin{cases} {\rm esssup} (u),& s = 0, \hfill \\ \mathop {\inf }\limits_t \left\{ {t:{\mu _u}(t) < s} \right\}, & s > 0. \hfill \\ \end{cases}

Essentially, u^\sharp is just the inverse function of the distribution function \mu_u of u. The following properties of the decreasing rearrangement are immediate from its definition.

Proposition 1. Let u : \Omega \to \mathbb R^N where \Omega \subset \mathbb R^N is bounded. Then u^\sharp is a nonincreasing and left-continuous function.

Proposition 2. The mapping u \mapsto u^\sharp is non-decreasing, i.e. if u\leqslant v in the sense that u(x) \leqslant v(x) for all x, where u and v are real-valued functions on \Omega then u^\sharp \leqslant v^\sharp.

We now see that u^\sharp is indeed a rearrangement of u.

Proposition 3. The function u : \Omega \to \mathbb R and u^\sharp : [0,|\Omega|] \to \mathbb R are equimeasurable (i.e. they have the same distribution function), i.e. for all t

\displaystyle |\{u >t\}|=|\{u^\sharp >t\}|.

Immediately one has

Corollary 1. With the preceding notations, we have

\displaystyle\begin{gathered} \left| {\left\{ {u > t} \right\}} \right| = \left| {\left\{ {{u^\sharp } > t} \right\}} \right|, \hfill \\ \left| {\left\{ {u \geqslant t} \right\}} \right| = \left| {\left\{ {{u^\sharp } \geqslant t} \right\}} \right|, \hfill \\ \left| {\left\{ {u < t} \right\}} \right| = \left| {\left\{ {{u^\sharp } < t} \right\}} \right|, \hfill \\ \left| {\left\{ {u \leqslant t} \right\}} \right| = \left| {\left\{ {{u^\sharp } \leqslant t} \right\}} \right|. \hfill \\ \end{gathered}

It is worth recalling from the theory of L^p-functions that

\displaystyle\int_\Omega {{{\left| {u(x)} \right|}^p}dx} = p\int_0^\infty {{t^{p - 1}}{\mu _u}(t)dt}

which gives

Corollary 2. If u \geqslant 0 and if u\in L^p(\Omega) for 1 \leqslant p \leqslant \infty then u^\sharp \in L^p((0, |\Omega|)) and

\displaystyle \|u\|_p=\|u^\sharp\|_p.

In literature, we have the following general and powerful result

Theorem. Let u :\Omega \to \mathbb R be measurable. Let F:\mathbb R \to \mathbb R be a non-negative Borel measurable function. Then

\displaystyle\int_\Omega {F(u(x))dx} = \int_0^{|\Omega |} {F({u^\sharp }(s))ds} .

We will end this first entry by providing several rearrangement inequalities.

Theorem (L^p continuity). Let 1 \leqslant p \leqslant \infty. The mapping u\mapsto u^\sharp is continuous from L^p(\Omega) into L^p((0,|\Omega|)).

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 f\in L^p(\Omega) and g \in L^q(\Omega) where \frac{1}{p}+\frac{1}{q}=1 and 1 \leqslant p,q\leqslant \infty. Then

\displaystyle\int_\Omega {f(x)g(x)dx} \leqslant \int_0^{|\Omega |} {{f^\sharp }(s){g^\sharp }(s)ds}.

The proof of the Hardy-Littlewood inequality relies on a density argument and the following useful identity

Lemma. Let f,g : \Omega \to \mathbb R with g integrable over \Omega. Let a \leqslant f\leqslant b\leqslant +\infty with a \in \mathbb R. Then

\displaystyle\int_\Omega {f(x)g(x)dx} = a\int_\Omega {g(x)dx} + \int_a^b {\left( {\int_{\left\{ {f > t} \right\}} {g(x)dx} } \right)dt} .

We now see the fact that the rearrangement mapping is non-expansive between the relevant L^p spaces.

Theorem (non-expansive property). Let f,g \in L^p(\Omega) where 1 \leqslant p\leqslant \infty. Then

\displaystyle {\left\| {{f^\sharp } - {g^\sharp }} \right\|_p} \leqslant {\left\| {f - g} \right\|_p}.

The proof of this theorem relies on a technique called truncation technique.

Similarly, we have the following variant

Definition (Increasing rearrangement). Let \Omega \subset \mathbb R^N be bounded and let u :\Omega \to  \mathbb R be a measurable function. Then the (unidimensional) increasing rearrangement of u, denoted by u_\sharp, is defined on [0, |\Omega|] by

\displaystyle {u_\sharp }(s) = \begin{cases} {\rm esssup}  (u), & s = 0, \hfill \\ \mathop {\inf }\limits_t \left\{ t: \{u < t\} < s\right\}, & s > 0. \hfill \\ \end{cases}

Obviously,

u^\sharp (s)=u_\sharp(|\Omega|-s)

and

u^\sharp (s)=-(-u)_\sharp

almost everywhere.

See also: Symmetrization And Applications (Series in Analysis) by S. Kesavan

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