Bounded mean oscillation

Summary

A locally integrable function has bounded mean oscillation if its average deviation from its own mean, over every cube, is bounded by one constant. The space BMO of such functions contains $L^\infty$ and also some unbounded functions such as $\log \abs{x}$. It is the dual of the real Hardy space $H^1$, by Fefferman's theorem. It is the natural replacement for $L^\infty$ in the theory of singular integrals.