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2016 Sharp weighted norm estimates beyond Calderón–Zygmund theory
Frédéric Bernicot, Dorothee Frey, Stefanie Petermichl
Anal. PDE 9(5): 1079-1113 (2016). DOI: 10.2140/apde.2016.9.1079


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.


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Frédéric Bernicot. Dorothee Frey. Stefanie Petermichl. "Sharp weighted norm estimates beyond Calderón–Zygmund theory." Anal. PDE 9 (5) 1079 - 1113, 2016.


Received: 5 October 2015; Revised: 19 February 2016; Accepted: 30 March 2016; Published: 2016
First available in Project Euclid: 12 December 2017

zbMATH: 1344.42009
MathSciNet: MR3531367
Digital Object Identifier: 10.2140/apde.2016.9.1079

Primary: 42B20, 58J35

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.9 • No. 5 • 2016
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