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Potential Analysis

, Volume 47, Issue 2, pp 169–187 | Cite as

Weak Type (1,1) Behavior for the Maximal Operator with L 1-Dini Kernel

  • Yong Ding
  • Xudong LaiEmail author
Article

Abstract

Let M Ω be the maximal operator with homogeneous kernel Ω. In the present paper, we show that if Ω satisfies the L 1-Dini condition on 𝕊 n−1, then the following weak type (1,1) behaviors
$$\lim\limits _{\lambda \rightarrow 0_{+}}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})=\frac {1}{n} \|\Omega \|_{1} \|f\|_{1},$$
$$\sup\limits_{\lambda >0}\lambda m(\{x\in \mathbb {R}^{n}:M_{\Omega } f(x)>\lambda \})\lesssim {\bigg ((\log n)\|\Omega \|_{1}+{\int }_{0}^{1/n}\frac {\tilde {\omega }_{1}(\delta )}{\delta }d\delta \bigg )}\|f\|_{1}$$

hold for the maximal operator M Ω and \(f\in L^{1}(\mathbb {R}^{n})\), here \(\tilde {\omega }_{1}\) denotes the L 1 integral modulus of continuity of Ω defined by translation in \(\mathbb {R}^{n}\).

Keywords

Limiting behavior Weak type Maximal operator Homogeneous kernel 

Mathematics Subject Classification (2010)

42B25 

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Copyright information

© Springer Science+Business Media Dordrecht 2017

Authors and Affiliations

  1. 1.School of Mathematical Sciences, Laboratory of Mathematics and Complex Systems (BNU), Ministry of EducationBeijing Normal UniversityBeijingPeople’s Republic of China

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