Vanishing mean oscillation

Remarks

VMO is a closed subspace of BMO, and Sarason showed that it is the closure in BMO of the uniformly continuous functions in BMO. Some authors use a smaller space on $\R^n$, also called VMO: the closure in BMO of the continuous functions with compact support. The dual space of this smaller space is the real Hardy space $H^1$.

Example: every uniformly continuous $f$ in BMO is in VMO. If $\abs{f(x) - f(y)} \le \varepsilon$ whenever $\abs{x - y} \le \eta$, then on a cube $Q$ of diameter at most $\eta$ one has $\abs{f(x) - f_Q} \le \varepsilon$ for $x \in Q$, so $\omega_f(\delta) \le \varepsilon$ for $\delta$ small.

Example with a singularity: if $n \ge 2$ and $\nabla f \in L^n(\R^n)$, then the Poincaré inequality on cubes gives $\frac{1}{\abs{Q}} \int_Q \abs{f - f_Q} \le C \left( \int_Q \abs{\nabla f}^n \right)^{1/n}$, which tends to $0$ uniformly as $\abs{Q} \to 0$. So such $f$ are in VMO, although they can be unbounded.

Non-example: $f(x) = \log \abs{x}$ on $\R$ is in BMO but not in VMO. For $r > 0$, $f(r x) = f(x) + \log r$, so the mean oscillation of $f$ on $[-r, r]$ equals its mean oscillation on $[-1, 1]$, which is a fixed positive number. Hence $\omega_f(\delta)$ does not tend to $0$.