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We investigate the small constant case of a characterization of $A_\infty$weights due to Fefferman, Kenig and Pipher. In their work, Fefferman, Kenig andPipher bound the logarithm of the $A_\infty$ constant by the Carleson norm of ameasure built out of the heat extension, up to a multiplicative and additiveconstant (as well as the converse). We prove, qualitatively, that when one ofthese quantities is small so is the other. In fact, we show that thesequantities are bounded by a constant times the square root of the other,provided at least one of them is sufficiently small.  We also give an application of our result to the study of elliptic measuresassociated to elliptic operators with coefficients satisfying the``Dahlberg-Kenig-Pipher" condition. We suspect that the square root dependencein the bound used in this application is sharp and give some justification forthis in the last section.

Carleson Conditions for Weights: The quantitative small constant case