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A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on \({\mathbb {R}}^2\)

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Abstract

Let \(\varphi _1, \gamma \) be nondecreasing functions on \([0,\infty ) \), \(\varphi _2\) be a quasi-convex function and \(M^+f\) be the one-sided Hardy–Littlewood maximal function on \(\mathbb {R}^2\). In this paper, we give a characterization theorem for a weighted weak type inequality of the form

$$\begin{aligned} \varphi _1(\lambda )\omega (\{x \in \mathbb {R}^2: M^+f(x)> \lambda \}) \le C\int _{\mathbb {R}^2}\varphi _2 \left( C\frac{|f(x)|\varrho (x)}{\gamma (\lambda )} \right) \sigma (x)dx, \end{aligned}$$

which generalizes and unifies some known results.

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References

  1. Bagby, R.J.: Weak bounds for the maximal function in weighted Orlicz spaces. Stud. Math. 95, 195–204 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berkovits, L.: Parabolic Muckenhoupt weights in the Euclidean space. J. Math. Anal. Appl. 379(2), 524–537 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bloom, S., Kerman, R.: Weighted Orlicz space integral inequalities for the Hardy–Littlewood maximal operator. Studia Math. 2(110), 149–167 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  4. Ding, S., Ren, Y.B.: Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spaces. Acta Math. Hungar. 162(2), 677–689 (2020)

  5. Forzani, L., Martín-Reyes, F.J., Ombrosi, S.: Weighted inequalities for the two-dimensional one-sided Hardy–Littlewood maximal function. Trans. Am. Math. Soc. 363(4), 1699–1719 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gallardo, D.: Weighted weak type integral inequalities for the Hardy–Littlewood maximal operator. Israel J. Math. 67(1), 95–108 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  7. Ghosh, A., Mohanty, P.: Weighted inequalities for higher dimensional one-sided Hardy–Littlewood maximal function in Orlicz spaces. Expo. Math. 40(1), 23–44 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  8. Gogatishvili, A., Kokilashvili, V.: Criteria of weighted inequalities in Orlicz classes for maximal functions defined on homogeneous type spaces. Georgian Math. J. 1(6), 641–673 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  9. Gogatishvili, A., Kokilashvili, V.: Necessary and sufficient conditions for weighted Orlicz class inequalities for maximal functions and singular integrals. I. Georgian Math. J. 2, 361–384 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kokilashvili, V., Krbec, M.: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific Publishing, Singapore (1991)

    Book  MATH  Google Scholar 

  11. Lai, Q.: Two-weight mixed \(\Phi \)-inequalities for the one-sided maximal function. Stud. Math. 115(1), 1–22 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  12. Lai, Q.: A note on the weighted norm inequality for the one-sided maximal operator. Proc. Am. Math. Soc. 124(2), 527–537 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  13. Lerner, A.K., Ombrosi, S.: A boundedness criterion for general maximal operators, pp. 53–71. Publicacions Matemàtiques (2010)

  14. Lorente, M., Martín-Reyes, F.J.: The one-sided dyadic Hardy–Littlewood maximal operator. Proc. R. Soc. Edinb. Sec. A Math. 144(5), 991–1006 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  15. Martín-Reyes, F.J.: New proofs of weighted inequalities for the one-sided Hardy–Littlewood maximal functions. Proc. Am. Math. Soc. 117(3), 691–698 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Martín-Reyes, F.J., Alberto, D.: Sharp weighted bounds for one-sided maximal operators. Coll. Math. 66(2), 161–174 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  17. Martín–Reyes, F.J., Ortega Salvador, P., De la Torre, A.: Weighted inequalities for the one–sided maximal functions. Trans. Am. Math. Soc. 319(2), 517–534 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  18. Muckenhoupt, B.: Weighted norm inequalities for the Hardy maximal function. Trans. Am. Math. Soc. 165, 207–226 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ombrosi, S.: Weak weighted inequalities for a dyadic one-sided maximal function in \({\mathbb{R} }^n\). Proc. Am. Math. Soc. 133(6), 1769–1775 (2005)

    Article  MATH  Google Scholar 

  20. Ortega Salvador, P., Pick, L.: Two-weight weak and extra-weak type inequalities for the one-sided maximal operator. Proc. Royal. Soc. Edinbur. Sec. A Math. 123(06), 1109–1118 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  21. Pick, L.: Two weight weak type maximal inequalities in Orlicz classes. Studia Math. 3(100), 207–218 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ren, Y.B., Ding, S.: Necessary and sufficient conditions for the two-weight weak type maximal inequality in Orlicz class. Czech. Math. J. 72(147), 79–85 (2022)

    Article  MathSciNet  MATH  Google Scholar 

  23. Riveros, M.S., Torre, A.: Norm inequalities relating one-sided singular integrals and the one-sided maximal function. J. Austr. Math. Soc. 69(3), 403–414 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Salvador, P.O.: Weighted inequalities for one-sided maximal functions in Orlicz spaces. Studia Math. 131(2), 101–114 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Salvador, P.O., Torreblanca, C.R.: Weighted inequalities for the one-sided geometric maximal operators. Math. Nach. 284(12), 1515–1522 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Sawyer, E.: Weighted inequalities for the one-sided Hardy–Littlewood maximal functions. Trans. Am. Math. Soc. 297(1), 53–61 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  27. Wang, J., Ren, Y.B., Zhang, E.X.: A weighted weak type inequality for the one-sided maximal operator. Ukr. Math. J. 75(5), 712–720 (2023)

    Google Scholar 

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Acknowledgements

The first author was supported by the National Natural Science Foundation of China (Grant No.12101193). The second author was supported by the National Natural Science Foundation of China (Grant No.11871195).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, People’s Republic of China

    Erxin Zhang

  2. School of Mathematics and Statistics, Yancheng Teachers University, Yancheng, 224002, People’s Republic of China

    Yanbo Ren

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  1. Erxin Zhang
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  2. Yanbo Ren
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Correspondence to Erxin Zhang.

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Communicated by Yoshihiro Sawano.

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Zhang, E., Ren, Y. A Unified Version of Weighted Weak Type Inequalities for One-Sided Maximal Function on \({\mathbb {R}}^2\). Bull. Malays. Math. Sci. Soc. 46, 122 (2023). https://doi.org/10.1007/s40840-023-01511-4

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  • DOI: https://doi.org/10.1007/s40840-023-01511-4

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