A sharp estimate for Muckenhoupt class A∞ and BMO

A classical fact in the weighted theory asserts that a weight w belongs to the Muckenhoupt class A∞ if and only if its logarithm log w is a function of bounded mean oscillation. We prove a sharp quantitative version of this fact in dimension one: for a weight w dened on some interval J ⊂ R, we provide best lower and upper bounds for the BMO norm of log w in terms of A∞ characteristics of w. The proof rests on the precise evaluation of associated Bellman functions.