Bounded mean oscillation

Remarks

$\norm{\cdot}_{\mathrm{BMO}}$ vanishes exactly on the constant functions, so it is a norm on BMO modulo constants, which is a Banach space. The same definition with arcs of the unit circle in place of cubes gives $\mathrm{BMO}(\mathbb{T})$. Every bounded function lies in BMO with $\norm{f}_{\mathrm{BMO}} \le 2 \sup \abs{f}$. The John–Nirenberg inequality says that a BMO function is exponentially integrable on each cube after subtracting its mean.

Example: $f(x) = \log \abs{x}$ on $\R$ lies in BMO. For $\lambda > 0$ one has $f(\lambda x) = f(x) + \log \lambda$, and adding a constant does not change the mean oscillation. So it suffices to bound the mean oscillation on intervals of length $1$, where a direct estimate works.

Non-example: $f(x) = x$ on $\R$. On $Q = [0, L]$ one has $f_Q = L/2$ and $\frac{1}{L} \int_0^L \abs{x - L/2} \, dx = L/4$, which is unbounded in $L$.