This is an announcement for the paper "The foundational inequalities of D.L. Burkholder and some of their ramifications" by Rodrigo Banuelos.

Abstract: This paper present an overview of some of the applications of the martingale transform inequalities of D.L.~Burkholder to $L^p$-bounds for singular integrals concentrating on $L^p$-bounds for the Hilbert, Riesz, Beurling-Ahlfors transforms and other multipliers obtained by projections (conditional expectations) of transformations of stochastic integrals. The aim is to obtain optimal, or near optimal, bounds in these inequalities. Connections to other areas of mathematics where these inequalities and the techniques to prove them have become of considerable interest in recent years, are also discussed.

Archive classification: math.PR math.AP math.FA

Submitted from: banuelos@math.purdue.edu

The paper may be downloaded from the archive by web browser from URL

http://front.math.ucdavis.edu/1012.4850

or

http://arXiv.org/abs/1012.4850