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Math Help - Product of Lebesgue integrable functions...

  1. #1
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    Product of Lebesgue integrable functions...

    Let f $\f \in {L^4}\left( R \right),g \in {L^5}\left( R \right)\ . Check the following:

    1)  fg$\fg \in {L^{20/9}}(R)\$

    2) $\[fg \notin {L^6}\left( R \right)\]$

    Where
    $\L^P} \buildrel \Delta \over = \left\{ {f:I \to R/f Lebesgue int} \right\}\

    Thanks...
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  2. #2
    Super Member girdav's Avatar
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    We have |f|^{\frac{20}9} \in L^{4\frac 9{20}} and |g|^{\frac{20}9} \in L^{5\frac 9{20}}. We use Hölder inequality (with p =4\frac 9{20} and q =5\frac 9{20}) and we get |fg|^{\frac{20}9}\in L^{\frac{20}{4\cdot 9}+\frac{20}{5\cdot 9}}=L^1.

    For the second I don't understand : what about f(x)=g(x) =e^{-|x|}?
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