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

Abstract

Sharp inequalities between weight bounds (from the doubling, Ap, and reverse Holder 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 Holder 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.