Exotic Calderón–Zygmund Operators

We study singular integral operators with kernels that are more singular than standard Calderón–Zygmund kernels, but less singular than bi-parameter product Calderón–Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmu...

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Bibliographic Details
Published in:The Journal of geometric analysis 2023-05, Vol.33 (5), Article 157
Main Authors: Hytönen, Tuomas, Li, Kangwei, Martikainen, Henri, Vuorinen, Emil
Format: Article
Language:English
Subjects:
Abstract Harmonic Analysis
Commutators
Convex and Discrete Geometry
Differential Geometry
Dynamical Systems and Ergodic Theory
Estimates
Fourier Analysis
Geometry
Global Analysis and Analysis on Manifolds
Kernels
Mathematics
Mathematics and Statistics
Operators
Parameter estimation
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Summary:We study singular integral operators with kernels that are more singular than standard Calderón–Zygmund kernels, but less singular than bi-parameter product Calderón–Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T 1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calderón–Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.
ISSN:1050-6926
1559-002X
DOI:10.1007/s12220-023-01216-x