Ideal weights: Asymptotically optimal versions of doubling, absolute continuity, and bounded mean oscillation

Sharp inequalities between weight bounds (from the doubling, A_p, and reverse Hölder conditions) and the BMO norm are obtained when the former are near their optimal values. In particular, the BMO norm of the logarithm of a weight is controlled by the square root of the logarithm of its A_∞ bound. These estimates lead to a systematic development of asymptotically sharp higher integrability results for reverse Hölder weights and extend Coifman and Fefferman's formulation of the A_∞ condition as an equivalence relation on doubling measures to the setting in which all bounds become optimal over small scales.