The Martingale Structure of the Beurling-ahlfors Transform

The Beurling-Ahlfors operator reveals a rich structure through its representation as a martingale transform. Using elementary linear algebra and martingale inequalities, we obtain new information on this operator. In particular, Essén-type inequalities are proved for the complex Beurling-Ahlfors operator and its generalization to higher dimensions. Moreover, a new estimate of their norms is obtained for dimensions n ≥ 3. Finally, we discuss a purely analytic approach to further investigate these norms that is suggested by the probabilistic method presented here. §0. Introduction. The complex Beurling-Ahlfors transform and its generalization to higher dimensions have been used to study properties of quasiregular mappings for several years. It is of particular interest to identify the L-norms of these operators. These norms are directly connected to the regularity of quasiregular mappings, as well as conditions for a closed set to be removable under such maps. Identification of the norms would also have implications for the existence of minimizers of conformally invariant energy functionals and regularity of solutions to the generalized Beltrami system. See [IM1], [IM2], [IMNS] and [IL] for further discussion of these operators and their relationship to quasiregular mappings. The Beurling-Ahlfors operator in several dimensions was introduced by Donaldson and Sullivan as the “signature operator” in their paper Quasiconformal 4manifolds [DS]. The first systematic study of this operator as a Calderón-Zygmund singular integral in higher dimensions was made by Iwaniec and Martin in [IM1]. There it is observed that the operator possesses a rich invariance structure. The probabilistic study of the Beurling-Ahlfors transform was initiated in [BW] and [L]. Using some new sharp martingale inequalities inspired by the recent work of ∗,∗∗Research of both authors supported in part by the NSF under grant DMS9400854