Ngô Quốc Anh

May 14, 2010

Symmetrization: Schwarz symmetrization

Filed under: Giải tích 8 (MA5206) — Tags: — Ngô Quốc Anh @ 15:34

Given a measurable subset E \subset \mathbb R^N, we denote its N-dimensional Lebesgue measure by |E|. We will denote by E^\star the open ball centered at the origin and having the same measure as E, i.e. |E^\star|=|E|. The norm of vector x \in \mathbb R^n will be denoted by |x|. Finally, we will denote by \omega_N the volume of the unit ball in \mathbb R^N. It is worth recalling that

\displaystyle \omega_N=\frac{\pi^\frac{N}{2}}{\Gamma \left(\frac{N}{2}+1\right)}

where \Gamma us the usual gamma function.

Definition (Schwarz symmetrization). Let \Omega \subset \mathbb R^N be a bounded domain. Let u : \Omega \to \mathbb R be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and decreasing rearrangement) is the function u^\star : \Omega^\star \to \mathbb R defined by

u^\star(x)=u^\sharp (\omega_N|x|^N), \quad x \in \Omega^\star.

Observe that if R is the radius of \Omega^\star, then

\displaystyle\begin{gathered} \int_{{\Omega ^ \star }} {{u^ \star }(x)dx} = \int_{{\Omega ^ \star }} {{u^\sharp }({\omega _N}{{\left| x \right|}^N})dx} \hfill \\ \qquad= \int_0^R {{u^\sharp }({\omega _N}{{\left| x \right|}^N})N{\omega _N}{\tau ^{N - 1}}d\tau } \hfill \\ \qquad= \int_0^{|{\Omega ^ \star }|} {{u^\sharp }(s)ds} \hfill \\ \qquad= \int_0^{|\Omega |} {{u^\sharp }(s)ds} . \hfill \\ \end{gathered}

We obviously have the following properties of Schwarz symmetrization

  • u^\star is radially symmetric and decreasing.
  • u, u^\sharp, u^\star are all equi-measurable.
  • If F: \mathbb R \to\mathbb R is a Borel measurable function such that either F \geqslant 0 or F(u) \in L^1(\Omega) then

    \displaystyle\int_{{\Omega ^ \star }} {F({u^ \star }(x))dx} = \int_\Omega {F(u(x))dx}.

    In particular, u and u^\star have the same L^p-norm and

    \displaystyle\int_{{\Omega ^ \star }} {{u^ \star }(x)dx} = \int_\Omega {u(x)dx}.

    when u is integrable over \Omega.

  • If \psi : \mathbb R \to \mathbb R is a non-decreasing function, then
    \displaystyle {(\psi (u))^ \star } = \psi ({u^ \star }).
  • The mapping u : \mapsto u^\star is a non-expansive mapping from L^p(\Omega) into L^p(\Omega^\star for all 1 \leqslant p \leqslant \infty.
  • If E \subset \Omega is a measurable subset, then

    \displaystyle\int_E {u(x)dx} \leqslant \int_0^{|E|} {{u^\sharp }(s)ds} = \int_{{E^ \star }} {{u^ \star }(x)dx}.

    Equality occurs iff

    \displaystyle (u\big|_E)^\star=u^\star\big|_{E^\star}.

  • The Hardy-Littlewood inequality holds, i.e.

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

    for any 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.

Similarly, we have the following variant.

Definition (Schwarz symmetrization). Let \Omega \subset \mathbb R^N be a bounded domain. Let u : \Omega  \to \mathbb R be a measurable function. Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function u_\star : \Omega^\star \to \mathbb  R defined by

u_\star(x)=u_\sharp (\omega_N|x|^N), \quad x \in \Omega^\star.

Now if we replace a bounded domain \Omega by the whole space \mathbb R^N, we have the following variant

Definition (Schwarz symmetrization for the whole space). Let f : \mathbb R^N \to \mathbb R be a measurable function which is vanishing at infinity in the sense that for any t>0

\displaystyle | \{|f|>t\}| < \infty.

Then, its Schwarz symmetrization (or the spherically symmetric and increasing rearrangement) is the function f^\star : \mathbb R^N \to \mathbb   R defined by

\displaystyle {f^ \star }(x) = \int_0^\infty {{\chi _{{{\left\{ {|f| > t} \right\}}^ \star }}}(x)dt}.

Interestingly, f^\star 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

\displaystyle\int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {f(x)g(x - y)h(y)dxdy} } \leqslant \int_{{\mathbb{R}^N}} {\int_{{\mathbb{R}^N}} {{f^ \star }(x){g^ \star }(x - y){h^ \star }(y)dxdy} }

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

See also:

3 Comments »

  1. 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

  2. Thanks Fab. A short but interesting note.

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

  3. 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


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