We investigate the small constant case of a characterization of A∞ weights due to Fefferman, Kenig and Pipher (1991): the logarithm of the A∞-constant is controlled by the Carleson norm of a measure built out of the heat extension, up to a multiplicative and additive constant. It was left open whether one of these quantities being small implies the same for the other. We hope to show (the use of the word « hope » is essential here; stay tuned) that these quantities are bounded by a constant times the square root of the other, provided at least one of them is sufficiently small. Such results have applications to the study of elliptic measures associated to elliptic operators with coefficients satisfying the « Dahlberg-Kenig-Pipher »  condition.

Based on joint work with S. Bortz and O. Saari.