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The question is as in the above.

In all literature, I only find that on $[0,1]$ with the Lebesgue measure, $\lVert \cdot \rVert_q \leq \lVert \cdot \rVert_p$ for $p>q$.

(I deleted the last question because it was poorly worded...) but in the last question, what I wanted to know is the possibility of reversing this inequality.

Namely, for a given pair $p>q$ and all $f \in L^q[0,1]$ with $\lVert f \rVert_q \geq 1$, is it possible to have some very large but "fixed" $n$ such that \begin{equation} \lVert f \rVert_p^{1/n} \leq \lVert f \rVert_q \end{equation}

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    $\begingroup$ No. Take $f = \varepsilon^{-1/q} 1_{[0,\varepsilon]}$ for a small $\varepsilon>0$. (More generally, step functions tend to be excellent test cases for these sorts of inequalities.) $\endgroup$
    – Terry Tao
    May 23, 2023 at 17:42
  • $\begingroup$ Oh, I see. What if I change the condition to $\lVert f \rVert_q >1$ instead? $\endgroup$
    – Isaac
    May 23, 2023 at 17:44
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    $\begingroup$ Now take $f = 2\varepsilon^{-1/q} 1_{[0,\varepsilon]}$. $\endgroup$
    – Terry Tao
    May 23, 2023 at 17:45

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