FUNCTIONS OF BOUNDED MEAN OSCILLATION

$BMO$, the space of functions of bounded mean oscillation, was first introduced by F. John and L. Nirenberg in 1961. It became a focus of attention when C. Fefferman proved that $BMO$ is the dual of the (real) Hardy space $H^1$ in 1971. In the past 30 years, this space was studied extensively by many mathematicians. With the help of $BMO$, many phenomena can be characterized clearly. In this review we discuss the connections between $BMO$ functions, the sharp function operator, Carleson measures, atomic decompositions and commutator operators in $\bf{R}^n$. We strive to cover some of the main developments in the theory, including $BMO$ in a bounded Lipschitz domain in $\bf{R}^n$ and in the product space $\bf{R}\times \bf{R}$.