Bounded Mean Oscillation with respect to Banach function spaces

In the 1960s John and Nirenberg introduced the space of bounded mean oscillation functions $BMO$ in connection with differential equations. Since that time, and because of the diverse and direct relationship with other relevant objects in Harmonic Analysis, such as duality of Hardy spaces, upper endpoint estimates of Calderón-Zygmund operators, and the $L^p$ estimates of Commutators of those operators, $BMO$ spaces have been objective of much study. In this talk, we will discuss necessary and sufficient (geometric) conditions in a Banach function space $X$ in such a way that $BMO$ and $BMO_{X}$ are equivalent spaces. The new results that we will present in this talk are based on joint works with E. Lorist and A. Lerner.