A MARTINGALE STUDY OF THE BEURLING-AHLFORS TRANSFORM IN R(N)

We apply the theory of martingale transforms to study the Beurling-Ahl fors transform, S, in dimensions n greater than or equal to 2. This Op erator reveals a rich structure through its representation as a martin gale, and we obtain new results concerning the operator norm of S acti ng on the class of differential Terms having L(p) coefficients. In par ticular, we show that its norm is independent of the dimension when re stricted to k-forms and we present new ''Essen-type'' norm inequalitie s related to this martingale structure. Finally, we suggest a purely a nalytic method to further investigate these norms which up to now has been lacking. (C) 1997 Academic Press.