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Let $I=(0, 1)$. Suppose we are given $1<p<+\infty$ and a sequence of function $f_n$ which is bounded in $L^p(I)$. I was thinking.. if we also assume that $f_n$ converges to $0$ in $L^1(I)$, can we conclude that $f_n$ converges to $0$ in $L^r(I)$, at least for $1\leq r<p$?

Many thanks

Guido

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  • $\begingroup$ Do you mean that $\|f_n\|_p$ is bounded, or the functions $f_n$ are bounded (uniformly, presumably) and in $L^p$? $\endgroup$
    – copper.hat
    Aug 29, 2012 at 15:41
  • $\begingroup$ $\|f_n\|_p$ is bounded $\endgroup$ Aug 29, 2012 at 15:45

2 Answers 2

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$$ \|f_n\|_r^r\leqslant\|f_n\|_p^{p(r-1)/(p-1)}\cdot\|f_n\|_1^{(p-r)/(p-1)}\leqslant C\cdot\|f_n\|_1^{(p-r)/(p-1)}\underset{n\to\infty}{\longrightarrow}0 $$

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  • $\begingroup$ Very succinct... $\endgroup$
    – copper.hat
    Aug 29, 2012 at 16:58
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More generally (than the answer by did), given $1\le p,q<\infty$, any $r$ strictly between $p$ and $q$ can be written $$r=\frac{p}{s}+\frac{q}{s'},\qquad\frac{1}{s}+\frac{1}{s'}=1.$$ Then $$\def\abs#1{\lvert#1\rvert}\def\norm#1{\lVert#1\rVert} \int\abs{f}^r=\int\abs{f}^{p/s}\cdot\abs{f}^{q/s'} \le\norm{f^{p/s}}_s\cdot\norm{f^{q/s'}}_{s'} =\norm{f}_p^{p/s}\cdot\norm{f}_q^{q/s'}, $$ so you can generalize the problem (and did's solution) by replacing $1$, $p$ by $p$, $q$.

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