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Proving Bounded operator

  1. Apr 12, 2012 #1
    1. The problem statement, all variables and given/known data
    The operator T maps from [itex]L^p(-2,2)\rightarrow L^p(-2,2)[/itex] is defined [itex] (Tf)(x) = f(x) x[/itex]
    Show that the operator maps from L^p(-2,2) into the same.
    2. Relevant equations
    p is a natural from 1 to infinity.
    Holders inequality
    Substitution integrals
    3. The attempt at a solution
    I look at the following integral
    [itex] \int\limits_{-2}^{2} |f(x)x|^pdx = \int\limits_{-2}^{2}|f(x)x|^{p-1}|f(x)x|dx\leq
    \int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|f(x)|^rdx\right)^{1/r}\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right] = \\
    ||f(x)||_p\int\limits_{-2}^{2} |f(x)x|^{p-1}\left[\left(\int\limits_{-2}^{2}|x|^qdx\right)^{1/q}\right][/itex]
    And here im stuck
    Edit: maybe a substitution would do? I really need a hint.
    Last edited: Apr 12, 2012
  2. jcsd
  3. Apr 14, 2012 #2


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    I don't think you need anything fancy here: just notice that |x|<=2.
  4. Apr 14, 2012 #3
    You are right :) i made it. Tnx
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