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I'm reading a paper on the classical Gagliardo-Nirenberg interpolation inequality arXiv link and there is a inequality used $$ |v-\overline{v}|\le \left\Vert v' \right\Vert_{r,I} \ell^{1-\frac{1}{r}}, r\ge 1 $$ where $\overline{v}:=\frac{1}{\ell}\int_I v(x)dx$, $I$ is an interval on $R$, $v'$ is the derivative.

It looks quite simple, quite similar to the Hölder inequality, but where does the derivative come from? And I know that using the Poincaré inequality there will be a constant bound $C$, but then where does the term $\ell^{1-1/r}$ come from?

I think it should be a quite simple question but I am just stuck at it. Thanks for your comments!

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By the mean value theorem, $\bar v=v(t)$ for some $t\in I$. So, for all $x\in I$, $$|v(x)-\bar v|=|v(x)-v(t)| =\Big|\int_t^x v'\Big| \le\int_I|v'|\le\|v'\|_r\, \ell^{1-1/r};$$ the latter inequality is an instance of Hölder's inequality.

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  • $\begingroup$ Wow, it is quite clear. I have attemped to do like this but I tried like this: $|v-\overline{v}|=|\int_{x_1}^x \frac{dv}{dx}dx+v(x_1)-\overline{v}|\le \int |\frac{dv}{dx}|dx + |v(x_1)-\overline{v}| $, but it has the residue. $\endgroup$
    – Xeh Deng
    Commented Mar 7, 2022 at 18:31
  • $\begingroup$ The mean value theorem solves the problem in my mind. Thx!~ $\endgroup$
    – Xeh Deng
    Commented Mar 7, 2022 at 18:32
  • $\begingroup$ You are welcome! $\endgroup$
    – Iosif Pinelis
    Commented Mar 7, 2022 at 18:34

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