Functions with small BMO norm

We characterize the functions with 'small' bounded mean oscillation (BMO) norm by establishing the precise connection between the space BMO and class A∞ of Muckenhoupt weights. We prove that there exists a universal constant c∗2 such that ∥F∥ BMO < c∗2 if and only if exp f ∈ A2, where c∗2 is the sharp constant in the John and Nirenberg inequality. Similarly, in dimension one, we prove that ∥F∥ BMO < 1 if and only if f ∈ A1. As application we introduce a structure of metric space in A∞ and prove that the closed unit ball of A∞ is a Banach space.