# $f,g\in \mathrm{VMO}$ but $f\cdot g\notin \mathrm{VMO}$

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?

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