A rotation method which gives linear estimates for powers of the Ahlfors–Beurling operator
✍ Scribed by Oliver Dragičević; Stefanie Petermichl; Alexander Volberg
- Publisher
- Elsevier Science
- Year
- 2006
- Tongue
- English
- Weight
- 226 KB
- Volume
- 86
- Category
- Article
- ISSN
- 0021-7824
No coin nor oath required. For personal study only.
✦ Synopsis
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 Cnp * , p * = max{p, p p-1 }. We further improve this estimate to obtain the optimal behaviour of the L p norms in question.