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Sharp estimates of martingale transforms in higher dimensions and applications to the Ahlfors-Beurling operator

Oliver Dragicevic, Stefanie Petermichl, Alexander Volberg

math.FAarXiv:math/0606006

Abstract

The main aspiration of this note is to construct several different Haar-type systems in euclidean spaces of higher dimensions and prove sharp Lp bounds for the corresponding martingale transforms. In dimension one this was a result of Burkholder. The motivation for working in this direction is the search for Lp estimates of the Ahlfors-Beurling operator.

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