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A inequality

  1. Jul 13, 2013 #1
    For what class of functions we have:
    $$
    \int_{\Omega} [f(x)]^m dx \leq
    C\Bigr ( \int_{\Omega} f(x)dx\Bigr)^{m},
    $$
    where ##\Omega## is open bounded and ##f## is measurable on ##\Omega## and ##C,m>0##.
     
  2. jcsd
  3. Jul 13, 2013 #2

    micromass

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    For all ##m##?

    Well, take ##f## positive. You want ##\|f\|_m\leq C^{1/m}\|f\|_1##, for all ##m##. So by taking limits, we get

    [tex]\|f\|_\infty = \lim_{m\rightarrow +\infty} \|f\|_m\leq \lim_{m\rightarrow +\infty}C^{1/m}\|f\|_1 = \|f\|_1[/tex]

    In particular, you want the sup-norm to exist. This already forces your function to be bounded a.e.
     
    Last edited: Jul 13, 2013
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