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Lorentz space

  1. Sep 1, 2010 #1
    I am reading the definition in wiki ( nothing better at the moment)

    It seems too vague for me, namely what they call "rearrangement function" [tex]f^{*}[/tex]:

    [tex]f^{*}: [0, \infty) \rightarrow [0, \infty]; \\

    f^{*}(t) = \inf\{\alpha \in \mathbb{R}^{+}: d_f(\alpha) \leq t\}; \\

    d_f(\alpha) = \mu(\{x \in X : |f(x)| > \alpha\}).


    I am trying to put in words what is written. Is it right:

    first for a given [tex]t[/tex]we are looking for all [tex]\alpha[/tex]'s, so that [tex]d_f(\alpha) \leq t[/tex], where [tex]d_f(\alpha) [/tex] is basically a size of the area where [tex]|f(x)| > \alpha [/tex]? Then we take infinum via [tex] \alpha [/tex], so as a result there will be the smallest [tex]d[/tex]?

    Still I cannot imagine "geometrically" how is it?

    At last, I need just simpler difinition for the case when [tex]f[/tex] is real.

    Last edited: Sep 1, 2010
  2. jcsd
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