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Prove that $$\lim_{n\to \infty}\int_0^{n^{-\beta}}nf(x)dx=0,\forall f\in L^q[0,1], $$ where $\beta=\frac s{s-1}$, $1<s<q$.

I tried to use Hölder's inequality, but it doesn't work.

Could you please give some hints? Many thanks in advance!

EDIT: I asked this question because I am trying to prove a conlusion in this question. (To make it easier for anyone with the same question to search.)

Davide Giraudo
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shwsq
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    why do you think Holder doesn't work? if you apply it to $1/p+1/q=1$ you get the integral of $|f|^q$ on the given domain and that clearly does to zero as $n \to \infty$ since the integration domain shrinks to zero, while $f \in L^q$ and then you get a term $n^{1-\beta/p}$ and a simple computation shows that that term also goes to zero since $\beta/p>1$ – Conrad Nov 29 '24 at 17:15
  • Oh I didn't realize $p-\beta<0$, being stupid. Thank you very much! – shwsq Nov 29 '24 at 17:18
  • happy to be of help – Conrad Nov 29 '24 at 17:27

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In fact it is true in the case $\beta=q/(q-1)$. Indeed, applying Hölder's inequality with the exponents $p=q/(q-1)$ and $q$, and $g=n\mathbf{1}_{(0,n^{-\beta})}$, $h=\mathbf{1}_{(0,n^{-\beta})}f$, we get $$ \left\lvert \int_0^{n^{-\beta}}nf(x)dx\right\rvert\leqslant n\lVert \mathbf{1}_{(0,n^{-\beta})}\rVert_p\lVert f\mathbf{1}_{(0,n^{-\beta})}\rVert_q=\lVert f\mathbf{1}_{(0,n^{-\beta})}\rVert_q $$ which goes to $0$ by monotone convergence.

Davide Giraudo
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