A rotation method which gives linear [formula omitted] estimates for powers of the Ahlfors–Beurling operator

Abstract

In [O. Dragičević, A. Volberg, Sharp estimate of the Ahlfors–Beurling operator via averaging martingale transforms, Michigan Math. J. 51 (2) (2003) 415–435] the Ahlfors–Beurling operator T was represented as an average of two-dimensional martingale transforms. The same result can be proven for powers T n . Motivated by [T. Iwaniec, G. Martin, Riesz transforms and related singular integrals, J. Reine Angew. Math. 473 (1996) 25–57], we deduce from here that ‖ T n ‖ p are bounded from above by C n p ∗ , p ∗ = max { p , p p − 1 } . We further improve this estimate to obtain the optimal behaviour of the L p norms in question.