Sharp weighted norm estimates beyond Calderón–Zygmund theory

Bernicot, Frédéric; Frey, Dorothee; Petermichl, Stefanie

doi:10.2140/apde.2016.9.1079

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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

DOI: 10.2140/apde.2016.9.1079

Abstract

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 $(p_{0}, q_{0})$ with $1 \leq p_{0} < 2 < q_{0} \leq \infty$ . The norm estimates obtained are powers $α$ of the characteristic used by Auscher and Martell. The critical exponent in this case is $p = 1 + p_{0} ∕ q_{0}^{'}$ . We prove $α = 1 ∕ (p - p_{0})$ when $p_{0} < p \leq p$ and $α = (q_{0} - 1) ∕ (q_{0} - p)$ when $p \leq p < q_{0}$ . In particular, we are able to obtain the sharp $A_{2}$ 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.

Keywords

singular operators, weights

Mathematical Subject Classification 2010

Primary: 42B20, 58J35

Milestones

Received: 5 October 2015

Revised: 19 February 2016

Accepted: 30 March 2016

Published: 29 July 2016

Authors

Frédéric Bernicot
Laboratoire Jean Leray CNRS - Universite de Nantes 2, rue de la Houssinière 44322 Nantes Cedex 3 France

Dorothee Frey
Delft Institute of Applied Mathematics Delft University of Technology P.O. Box 5031 2600 GA Delft Netherlands

Stefanie Petermichl
Institut de Mathématiques de Toulouse Université Paul Sabatier 118 Route de Narbonne 31062 Toulouse Cedex 9 France

Vol. 9, No. 5, 2016