Journal of Operator Theory

Volume 65, Issue 2, Spring 2011 pp. 307-324.

New estimates for the Beurling-Ahlfors operator on differential forms

Authors: Stefanie Petermichl (1), Leonid Slavin (2), and Brett D. Wick (3)
Author institution: (1) Universite Paul Sabatier Institut de Mathematiques de Toulouse 118 route de Narbonne F-31062 Toulouse Cedex 9, France
(2) Department of Mathematical Sciences 839 Old Chemistry University of Cincinnati Cincinnati, OH 45221, U.S.A.
(3) School of Mathematics Georgia Institute of Technology 686 Cherry Street Atlanta, GA 30332-0160, U.S.A.

Summary: We establish new $L^p$ estimates for the norm of the generalized Beurling-Ahlfors transform $\mathcal{S}$ acting on form-valued functions. Namely, we prove that $\norm{\mathcal{S}}_{L^p(\R^n;\Lambda)\to L^p(\R^n;\Lambda)}\leqslant n(p^{*}-1)$ where $p^*=\max\{p, p/(p-1)\},$ thus extending the recent Nazarov-Volberg estimates to higher dimensions. The even-dimensional case has important implications for quasiconformal mappings. Some promising prospects for further improvement are discussed at the end.

Keywords: Beurling-Ahlfors operator, differential forms, heat extensions, Bellman functions

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