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, $\{u \geqslant t\}$, $\{u=t\}$ 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.