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The \(L^2\)-Boundedness of the Variational Calderón–Zygmund Operators

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Abstract

In this paper, we verify the \(L^2\)-boundedness for the jump functions and variations of Calderón–Zygmund singular integral operators with the underlying kernels satisfying

$$\begin{aligned} \int _{\varepsilon \le |x-y|\le N} K(x,y)\mathrm{{d}}y=\int _{\varepsilon \le |x-y|\le N}K(x,y)\mathrm{{d}}x=0\; \forall 0<\varepsilon \le N<\infty , \end{aligned}$$

in addition to some proper size and smooth conditions. This result should be the first general criteria for the variational inequalities for kernels not necessarily of convolution type. The \(L^2\)-boundedness assumption that we verified here is also the starting point of the related results on the (sharp) weighted norm inequalities appeared in many recent papers.

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Acknowledgements

The authors would like to express their gratitude to the referees for giving several valuable suggestions, which have greatly improved the exposition of the paper.

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Authors and Affiliations

  1. Department of Applied Mathematics, School of Mathematics and Physics, University of Science and Technology Beijing, Beijing, 100083, China

    Yanping Chen

  2. School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China

    Guixiang Hong

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  1. Yanping Chen
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  2. Guixiang Hong
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Correspondence to Yanping Chen.

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The project was in part supported by: Yanping Chen’s National Natural Science Foundation of China (# 11871096, # 11471033); Guixiang Hong’s National Natural Science Foundation of China (# 11601396).

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Chen, Y., Hong, G. The \(L^2\)-Boundedness of the Variational Calderón–Zygmund Operators. J Geom Anal 33, 73 (2023). https://doi.org/10.1007/s12220-022-01177-7

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