Vol. 9, No. 5, 2016

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Sharp weighted norm estimates beyond Calderón–Zygmund theory

Frédéric Bernicot, Dorothee Frey and Stefanie Petermichl

Vol. 9 (2016), No. 5, 1079–1113

We dominate nonintegral singular operators by adapted sparse operators and derive optimal norm estimates in weighted spaces. Our assumptions on the operators are minimal and our result applies to an array of situations, whose prototypes are Riesz transforms or multipliers, or paraproducts associated with a second-order elliptic operator. It also applies to such operators whose unweighted continuity is restricted to Lebesgue spaces with certain ranges of exponents (p0,q0) with 1 p0 < 2 < q0 . The norm estimates obtained are powers α of the characteristic used by Auscher and Martell. The critical exponent in this case is p = 1 + p0q0. We prove α = 1(p p0) when p0 < p p and α = (q0 1)(q0 p) when p p < q0. In particular, we are able to obtain the sharp A2 estimates for nonintegral singular operators which do not fit into the class of Calderón–Zygmund operators. These results are new even in Euclidean space and are the first ones for operators whose kernel does not satisfy any regularity estimate.

singular operators, weights
Mathematical Subject Classification 2010
Primary: 42B20, 58J35
Received: 5 October 2015
Revised: 19 February 2016
Accepted: 30 March 2016
Published: 29 July 2016
Frédéric Bernicot
Laboratoire Jean Leray
CNRS - Universite de Nantes
2, rue de la Houssinière
44322 Nantes Cedex 3
Dorothee Frey
Delft Institute of Applied Mathematics
Delft University of Technology
P.O. Box 5031
2600 GA Delft
Stefanie Petermichl
Institut de Mathématiques de Toulouse
Université Paul Sabatier
118 Route de Narbonne
31062 Toulouse Cedex 9