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Math Help - Lp space problem, possibly using Holder inequality

  1. #1
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    Lp space problem, possibly using Holder inequality

    Here's the problem statement:

    If f\in L_p for some 0<p<\infty, and every set of positive measure in X has measure at least m, show that f\in L_q for all p<q\leq\infty, with \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p.
    I have no problem showing that f\in L_q. It can also be shown that f\in L_{\infty}, in case that helps. However, I cannot show that the inequality

    \lVert f\rVert_q\leq m^{\frac{1}{q}-\frac{1}{p}}\lVert f\rVert_p

    holds. Any help would therefore be much appreciated !

    (This is exercise 1.3.12 from this book, p41.)
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  2. #2
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    Re: Lp space problem, possibly using Holder inequality

    Let X_a=\{ x\in X : |f(x)|>a\}. If you've proven that f\in L^\infty then let b=\| f\|_{\infty} then we have

    \int_X |f|^p \geq \int_{X_b} |f|^p \geq b^pm

    (Edit: To be formal you'd have to take X_{b-\varepsilon} but the bound on the left is independent of it and on the right it would look something like (b-\varepsilon)^pm \to b^pm)

    ie.  \| f\|_{\infty} \leq m^{-\frac{1}{p}} \| f\|_p. It's easy to prove by Hölder's inequality that f\in L^q and

    \| f\|_q \leq \| f\|^{\theta}_p \| f\|^{1-\theta}_{\infty} \qquad \frac{1}{q}=\frac{\theta}{p}

    now substitute the estimate for \| \cdot \|_\infty in this one and you're done.
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  3. #3
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    Re: Lp space problem, possibly using Holder inequality

    Very clever ! Thanks a bunch !
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