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Math Help - Lp Space Question

  1. #1
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    Lp Space Question

    Let (X, \mathcal{A}, \mu) be a measure space. Prove that if f belongs to L^4(\mu) and L^8(\mu) then it belongs to L^6(\mu) with ||f||_6 \leq ||f||_4^{\frac{1}{3}} \cdot ||f||_8^{\frac{2}{3}}.

    L^p(\mu) is contains equivalence classes where \int |f|^p d\mu < \infty.
    Also, ||f||_p=(\int |f|^p d\mu)^{\frac{1}{p}}.

    Attempt
    Let f \in L^4(\mu) and L^8(\mu). Then

     \int |f|^4 d\mu < \infty and  \int |f|^8 d\mu < \infty

    I don't see how to reach the conclusions now. I would appreciate any help on how to proceed. Thanks in advance.
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  2. #2
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    ||f||_6^6=||f^4 f^2||_1\leq ?

    Think of the French guy with the German for black...
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  3. #3
    Member mabruka's Avatar
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    CBS for the win
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