A Martingale Study of the Beurling–Ahlfors Transform inRn

We apply the theory of martingale transforms to study the Beurling–Ahlfors transform,S, in dimensionsn⩾2. This operator reveals a rich structure through its representation as a martingale, and we obtain new results concerning the operator norm ofSacting on the class of differential forms havingLpcoefficients. In particular, we show that its norm is independent of the dimension when restricted tok-forms and we present new “Essén-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.