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Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spaces

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Abstract

We obtain some new necessary and sufficient conditions for a multi-weight weak type maximal inequality of the form

$$\begin{aligned} \int _{\{ {x: \mathcal {M} f(x) > \lambda } \}} {\varphi (\lambda {\omega _1}(x))} {\omega _2}(x) \,d\mu \le {c} \int _X \varphi ({c}f(x){\omega _3}(x)){\omega _4}(x) \,d\mu \end{aligned}$$

in Orlicz classes, where \(\mathcal {M} f\) is a Hardy–Littlewood maximal function defined on homogeneous type spaces. Our main result extends some known results.

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References

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

    Article  MathSciNet  Google Scholar 

  2. Beltran, D.: Control of pseudodifferential operators by maximal functions via weighted inequalities. Trans. Amer. Math. Soc. 371, 3117–3143 (2019)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  4. Chen, W., Ho, K.P., Jiao, Y., Zhou, D.: Weighted mixed-norm inequality on Doob's maximal operator and John-Nirenberg inequalities in Banach function spaces. Acta Math. Hungar. 157, 408–433 (2019)

    Article  MathSciNet  Google Scholar 

  5. Cruz-Uribe, D., Shukla, P.: The boundedness of fractional maximal operators on variable Lebesgue spaces over spaces of homogeneous type. Studia Math. 242, 109–139 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  7. I. Genebashvili, A. Gogatishvili, V.Kokilashvili and M. Krbec, Weight Theory for Integral Transforms on Spaces of Homogeneous Type, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 92, Longman (Harlow, 1998)

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

    Article  MathSciNet  Google Scholar 

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

  10. A. Gogatishvili and J. S. Neves, Weighted norm inequalities for positive operators restricted on the cone of \(\lambda \)-quasiconcave functions, Proc. Roy. Soc. Edinb. Section A: Math., (2019), 1–23

  11. Gorosito, O., Kanashiro, A.M., Pradolini, G.: Characterizations of reverse weighted inequalities for maximal operators in Orlicz spaces and Stein's result. Acta Math. Hungar. 147, 354–367 (2015)

    Article  MathSciNet  Google Scholar 

  12. Kanashiro, A.M., Pradolini, G., Salinas, O.: Weighted modular estimates for a generalized maximal operator on spaces of homogeneous type. Collect. Math. 63, 147–164 (2012)

    Article  MathSciNet  Google Scholar 

  13. Karlovich, A.Y.: Maximal operators on variable Lebesgue spaces with weights related to oscillations of carleson curves. Math. Nachr. 283, 85–93 (2010)

    Article  MathSciNet  Google Scholar 

  14. Kita, H.: Weighted inequalities for iterated maximal functions in Orlicz spaces. Math. Nachr. 278, 1180–1189 (2005)

    Article  MathSciNet  Google Scholar 

  15. Kosz, D.: On relations between weak and strong type inequalities for maximal operators on non-doubling metric measure spaces. Publ. Mat. 62, 370–54 (2018)

    Article  MathSciNet  Google Scholar 

  16. Lin, H.B.: Sharp weighted bounds for the Hardy-Littlewood maximal operators on Musielak-Orlicz spaces. Arch. Math. 106, 275–284 (2016)

    Article  MathSciNet  Google Scholar 

  17. Mamedov, F.I., Zeren, Y.: Two-weight inequalities for the maximal operator in a Lebesgue space with variable exponent. J. Math. Sci. 173, 701–716 (2011)

    Article  MathSciNet  Google Scholar 

  18. Martell, J.M.: Fractional integrals, potential operators and two-weight, weak type norm inequalities on spaces of homogeneous type. J. Math. Anal. Appl. 294, 223–236 (2004)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  20. Mustafayev, R.: On weighted iterated Hardy-type inequalities. Positivity 22, 275–299 (2018)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. Qinsheng, L.: Two weight-inequalities for the Hardy operator, Hardy-Littlewood maximal operator, and fractional integrals. Proc. Amer. Math. Soc. 118, 129–142 (1993)

    Article  MathSciNet  Google Scholar 

  23. Y. Ren, A four-weight weak type maximal inequality for martingales, Acta Math. Sci. Ser. B (Engl. Ed.), 39 (2019), 413–419

  24. J. O. Strömberg and A. Torchinsky, Weighted Hardy Spaces, Springer (Berlin, Heidelberg, 1989)

  25. Trujillo-González, R.: Two-weight norm inequalities for fractional maximal operators on spaces of generalized homogeneous type. Periodica Math. Hungar. 44, 101–110 (2002)

    Article  MathSciNet  Google Scholar 

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Acknowledgement

We would like to express our sincere appreciations to the reviewers for their helpful suggestions.

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  1. School of Mathematics and Statistics, Henan University of Science and Technology, Luoyang, 471000, China

    S. Ding & Y. Ren

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  1. S. Ding
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  2. Y. Ren
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Correspondence to Y. Ren.

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Supported by the National Natural Science Foundation of China (Grant No. 11871195).

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Ding, S., Ren, Y. Criteria of a multi-weight weak type inequality in Orlicz classes for maximal functions defined on homogeneous type spaces. Acta Math. Hungar. 162, 677–689 (2020). https://doi.org/10.1007/s10474-020-01050-5

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