The Burkholder function and quasiconformal mappings

The Burkholder function, found in 1984 in the context of sharp martingale inequalities, turns out to be a “master function” which rules the sharp integrability properties of mappings in the plane. In this talk we will describe a proof of what we call the Burkholder area inequality, which is an optimal estimate for the Burkholder energy of quasiconformal maps in the spirit of the classical Grönwall-Bieberbach area formula. The talk is based on joint work with K. Astala, D. Faraco, A. Koski and J. Kristensen.