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

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

Oh! the little more, and how much it is! And the little less, and what worlds away!
Acknowledgements and Notes. Supported by the Max-Planck-Gesellschaft. This work is a revised form of part of the author's dissertation, which was written under Professor Carlos E. Kenig at the University of Chicago.