Quantitative weighted estimates for rough homogeneous singular integrals

We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2 ( w ), we obtain a bound that is quadratic in A 2 constant [ w ] A 2 . We do not know if this is sharp, b...

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Bibliographic Details
Published inIsrael journal of mathematics Vol. 218; no. 1; pp. 133 - 164
Main Authors Hytönen, Tuomas P., Roncal, Luz, Tapiola, Olli
Format Journal Article
LanguageEnglish
Published Jerusalem The Hebrew University Magnes Press 01.03.2017
Springer Nature B.V
Subjects
Algebra
Analysis
Applications of Mathematics
Decomposition
Group Theory and Generalizations
Integrals
Kernels
Mathematical and Computational Physics
Mathematics
Mathematics and Statistics
Operators (mathematics)
Series expansion
Theoretical
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Summary:We consider homogeneous singular kernels, whose angular part is bounded, but need not have any continuity. For the norm of the corresponding singular integral operators on the weighted space L 2 ( w ), we obtain a bound that is quadratic in A 2 constant [ w ] A 2 . We do not know if this is sharp, but it is the best known quantitative result for this class of operators. The proof relies on a classical decomposition of these operators into smooth pieces, for which we use a quantitative elaboration of Lacey's dyadic decomposition of Dini-continuous operators: the dependence of constants on the Dini norm of the kernels is crucial to control the summability of the series expansion of the rough operator. We conclude with applications and conjectures related to weighted bounds for powers of the Beurling transform.
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ISSN:0021-2172
1565-8511
DOI:10.1007/s11856-017-1462-6