Bounded mean oscillation

In harmonic analysis, a branch of mathematics, the space of functions of bounded mean oscillation (BMO) plays an important role.

A function u in $L^1_{\ell oc}(R^n)$ is said to be a BMO function if

$\sup_{x,r} \inf_a \frac{1}{|B_{x,r}|}\int_{B_{x,r}}|u(y)-a|\,dy=\|u\|_{BMO}<\infty.$

The infimum over a can be replaced by the choice of

$a=(u)_{B_{x,r}}=\frac{1}{|B_{x,r}|}\int_{B_{x,r}} u(y)\,dy.$