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Jump and Variational Inequalities for Rough Operators

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Abstract

In this paper, we systematically study jump and variational inequalities for rough operators, whose research have been initiated by Jones et al. More precisely, we show some jump and variational inequalities for the families \(\mathcal T:=\{T_\varepsilon \}_{\varepsilon >0}\) of truncated singular integrals and \(\mathcal M:=\{M_t\}_{t>0}\) of averaging operators with rough kernels, which are defined respectively by

$$\begin{aligned} T_\varepsilon f(x)=\int _{|y|>\varepsilon }\frac{\Omega (y')}{|y|^n}f(x-y)dy \end{aligned}$$

and

$$\begin{aligned} M_t f(x)=\frac{1}{t^n}\int _{|y|<t}\Omega (y')f(x-y)dy, \end{aligned}$$

where the kernel \(\Omega \) belongs to \(L\log ^+\!\!L(\mathbf S^{n-1})\) or \(H^1(\mathbf S^{n-1})\) or \(\mathcal {G}_\alpha (\mathbf S^{n-1})\) (the condition introduced by Grafakos and Stefanov). Some of our results are sharp in the sense that the underlying assumptions are the best known conditions for the boundedness of corresponding maximal operators.

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Acknowledgments

Authors would like to thank the anonymous referees for their valuable suggestions and corrections. The first author is supported by NSFC (11371057, 11471033, 11571160), SRFDP (20130003110003) and the Fundamental Research Funds for the Central Universities (2014KJJCA10). The second author is supported by MINECO: ICMAT Severo Ochoa project SEV-2011-0087. The third author is supported by NSFC (11371057, 11501169).

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Authors and Affiliations

  1. Laboratory of Mathematics and Complex Systems, School of Mathematical Sciences, Beijing Normal University (BNU), Ministry of Education of China, Beijing, 100875, China

    Yong Ding

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

    Guixiang Hong

  3. School of Mathematics and Information Science, Henan Polytechnic University, Jiaozuo, Henan, 454003, China

    Honghai Liu

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  1. Yong Ding
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  2. Guixiang Hong
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  3. Honghai Liu
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Correspondence to Honghai Liu.

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Communicated by Loukas Grafakos.

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Ding, Y., Hong, G. & Liu, H. Jump and Variational Inequalities for Rough Operators. J Fourier Anal Appl 23, 679–711 (2017). https://doi.org/10.1007/s00041-016-9484-8

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  • DOI: https://doi.org/10.1007/s00041-016-9484-8

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