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We say a function $ f\in L^1_{\mathrm{loc}}(\mathbb{R}) $ is in $\mathrm{BMO}(\mathbb{R})$ if

$$\|f\|_{\mathrm{BMO}}=\sup_{I}\frac{1}{|I|}\int\limits_I |f(y)-f_I|\, dy<\infty$$ for all intervals $I\subset\mathbb{R},$ where $f_I$ is the average value of $f$: $$f_I=\frac{1}{|I|}\int_I f(y)\, dy.$$

$f\in \mathrm{VMO}(\mathbb{R})$ if $f \in\mathrm{BMO}$ and $$\lim_{|I|\rightarrow 0}\frac{1}{|I|}\int_I|f(y)-f_I|\, dy\rightarrow 0. $$

I want to find two functions $ f $ and $ g $ such that $ f,g\in\mathrm{VMO}(\mathbb{R}) $ and $ f\cdot g\notin\mathrm{VMO}(\mathbb{R}) $. Can you give me some hints or references?

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    $\begingroup$ How about $f=g=\sin(1/x)\cdot x^a$ for $a\in [-1/2,-1)$? $\endgroup$
    – André Henriques
    Commented Mar 7, 2022 at 14:17
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    $\begingroup$ @AndréHenriques How is that in VMO? One can do $f(x)=g(x)=(\log(1+\frac1{|x|}))^{2/3}$ though. $\endgroup$
    – fedja
    Commented Mar 7, 2022 at 23:48

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