Rodrigo Bañuelos (Purdue University)

Title: Martingale transforms, the heart of the matter

Abstract: In 1966, D.L. Burkholder proved the boundedness of martingale transforms on $L^p$. While the result itself was of considerable importance, the ideas and techniques introduced in this paper became the building blocks for many subsequent results on martingales (including the celebrated Burkholder-Davis-Gundy inequalities) and their applications in many areas of probability and analysis. A major achievement of these ideas outside of probability was the solution in the early 70's by Burkholder, Gundy and Silverstein of a longstanding open problem of Hardy and Littlewood from 1930. In 1984, partly motivated by problems on the geometry of Banach spaces, Burkholder gave a new proof of his 1966 inequality which bypassed $L^2$ theory, introducing a novel and revolutionary new method for proving optimal inequalities in probability and harmonic analysis. This method, often referred to nowadays as the "Burkholder method" has had many applications in recent years in areas seemingly far removed from martingale theory.

The aim of this talk is to present a brief overview of martingale inequalities, including recent results of Adam Osękowski and the speaker obtained with the Burkholder method, and to discuss connections and applications to some open problems in harmonic analysis and calculus of variations.