A martingale study of the Beurling-Ahlfors transform in R(n)

We apply the theory of martingale transforms to study the Beurling-Ahlfors transform, S, in dimensions n greater than or equal to 2. This Operator reveals a rich structure through its representation as a martingale, and we obtain new results concerning the operator norm of S acting on the class of differential Terms having L(p) coefficients. In particular, we show that its norm is independent of the dimension when restricted to k-forms and we present new ''Essen-type'' norm inequalities related to this martingale structure. Finally, we suggest a purely analytic method to further investigate these norms which up to now has been lacking. (C) 1997 Academic Press.