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Problem of the Week #239 - Dec 27, 2016

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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
Here is this week's POTW:

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Let $1 < p < \infty$, and let $(f_n)$ be a sequence of real-valued functions in $\mathscr{L}^p(-\infty, \infty)$ which converges pointwise a.e. to zero. Show that if $\|f_n\|_p$ is uniformly bounded, then $(f_n)$ converges weakly to zero in $\mathscr{L}^p(-\infty,\infty)$.

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Remember to read the POTW submission guidelines to find out how to submit your answers!
 
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Euge

MHB Global Moderator
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Jun 20, 2014
1,892
No one answered this week's problem. You can read my solution below.



Let $q$ be the Hölder conjugate of $p$. Take $g\in \mathscr{L}^q(-\infty,\infty)$. By density of compactly supported functions in $\mathscr{L}^q$, it suffices to assume $g$ has compact support, and $\int f_n g \to 0$ as $n\to \infty$.

Suppose $g$ is supported on a compact set $K$. By Egorov's theorem, given $\epsilon > 0$ there exists a measurable subset $F$ of $K$ with $m(K\setminus F) < \epsilon$ such that $f_n \to f$ uniformly on $F$. Thus
$$\int_F \lvert f_ng\rvert \le \left(\sup_{x\in F} \lvert f_n(x)\rvert\right)\int_F\lvert g\rvert \quad \text{and}\quad \int_{K\setminus F} \lvert f_n g\rvert \le M\|g\|_\infty\epsilon^{1/q}$$
where $M = \sup_n\|f_n\|_p$. Hence
$$\varlimsup_{n\to \infty} \int \lvert f_n g\rvert \le 2M\|g\|_\infty \epsilon^{1/q}$$
Since $\epsilon$ was arbitary, the $\mathscr{L}^p$-weak limit of $f_n$ is zero.
 
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