The relation between BMO and bounded variation

Question

When studying BMO (bounded mean oscillation), it makes me think of its relation with bounded variation. These two both describe some kind of oscillation. However, it is quite different, e.x. bounded function is BMO, I want to know if there is relation with these two

Answer 1

$$BV(\mathbb R)\subsetneq BMO(\mathbb R)$$

The inclusion follows easily by noting that $BV$ functions are bounded. There are cheap examples showing the inclusion must be strict, like $x\mapsto x$ and $x\mapsto \sin(x).$ A more interesting example is the function defined a.e. by $f(x)=\log |x|.$ It's in $BMO(\mathbb R)$ (see this answer) even though its restriction to $(0,b)$ is not in $L^\infty(0,b)$ for any $b>0.$

For higher dimensions the two spaces are incomparable. The function $f(x_1,\dots,x_n)=\log|x_1|$ is in $BMO(\mathbb R^n)\setminus BV(\mathbb R^n).$ Let $\phi$ be some smooth function with $\phi(0)=1$ and rapid decay at infinity e.g. $\phi(r)=e^{-r^2}.$ Then the function defined by $f(x)=|x|^{-1/2}\phi(|x|)$ is in $W^{1,1}(\mathbb R^n)\subset BV(\mathbb R^n)$ but not in $BMO(\mathbb R^n).$