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

- Received:

DOI: 10.1007/BF02498222

- Cite this article as:
- Korey, M.B. The Journal of Fourier Analysis and Applications (1998) 4: 491. doi:10.1007/BF02498222

## Abstract

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.

### Math Subject Classifications

Primary 42B25, 26D15secondary 26B35### Keywords and Phrases

Doubling measurebounded mean oscillation*A*

_{∞}conditionreverse Hölder inequalityMuckenhoupt

*A*

_{p}conditionarithmetic-geometric inequality