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Thread: Lp Space Question

  1. #1
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    Lp Space Question

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

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

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

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

    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
    Member Focus's Avatar
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    $\displaystyle ||f||_6^6=||f^4 f^2||_1\leq ?$

    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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