Title: Heat extensions of doubling weights and A_infty

A_infty weights play a fundamental role in weighted inequalities for operators used in harmonic analysis. It is known that if w is an A_infty weight then log w in the space of bounded mean oscillation (BMO) and the converse is `almost’ true (up to taking a power of w). Motivated by this, R. Fefferman, Kenig and Pipher [FKP] showed a characterization of A_infty weights similar to the characterization of BMO by C. Fefferman and Stein.

Recently, with M. Egert and O. Saari, I showed a small constant version of [FKP]. When this is combined with some of my other work (with Toro and Zhao) we have a nice application to elliptic measure. We suspect that the application has a sharp power dependence; however, we left open some natural questions. In my talk, I will sketch the proofs of the my work with Egert and Saari, the application and discuss the open questions.