Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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TWO-WEIGHTED CONDITIONS AND CHARACTERIZATIONS FOR A CLASS OF MULTILINEAR FRACTIONAL NEW MAXIMAL OPERATORS

  • Rui Li (College of Mathematics and Statistics Northwest Normal University) ;
  • Shuangping Tao (College of Mathematics and Statistics Northwest Normal University)
  • Received : 2022.05.22
  • Accepted : 2022.09.26
  • Published : 2023.01.01
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Abstract

In this paper, two weight conditions are introduced and the multiple weighted strong and weak characterizations of the multilinear fractional new maximal operator 𝓜ϕ,β are established. Meanwhile, we introduce the ${\mathcal{S}}_{({\vec{p}},q),{\beta}}({\varphi})$ and $B_{({\vec{p}},q),{\beta}}({\varphi})$ conditions and obtain the characterization of two-weighted inequalities for 𝓜ϕ,β. Finally, the relationships of the conditions ${\mathcal{S}}_{({\vec{p}},q),{\beta}}({\varphi}),\,{\mathcal{A}}_{({\vec{p}},q),{\beta}}({\varphi})$ and $B_{({\vec{p}},q),{\beta}}({\varphi})$ and the characterization of the one-weight $A_{({\vec{p}},q),{\beta}}({\varphi})$ are given.

Keywords

Acknowledgement

This work was financially supported by the National Natural Science Foundation of China(Grant No. 11561062).

References

  1. M. Cao and Q. Xue, Characterization of two-weighted inequalities for multilinear fractional maximal operator, Nonlinear Anal. 130 (2016), 214-228. https://doi.org/10.1016/j.na.2015.10.004 
  2. M. Cao, Q. Xue, and K. Yabuta, On multilinear fractional strong maximal operator associated with rectangles and multiple weights, Rev. Mat. Iberoam. 33 (2017), no. 2, 555-572. https://doi.org/10.4171/RMI/949 
  3. W. Chen and W. Damian, Weighted estimates for the multisublinear maximal function, Rend. Circ. Mat. Palermo (2) 62 (2013), no. 3, 379-391. https://doi.org/10.1007/s12215-013-0131-9 
  4. X. Chen and Q. Xue, Weighted estimates for a class of multilinear fractional type operators, J. Math. Anal. Appl. 362 (2010), no. 2, 355-373. https://doi.org/10.1016/j.jmaa.2009.08.022 
  5. L. Grafakos, Classical Fourier analysis, second edition, Graduate Texts in Mathematics, 249, Springer, New York, 2008. 
  6. Y. P. Hu and Y. H. Cao, Two weight characterization of new maximal operators, Pure Appl. Math. J. 8 (2019), no. 3, 47-53. https://doi.org/10.11648/j.pamj.20190803.11 
  7. M. T. Lacey and K. Li, Two weight norm inequalities for the g function, Math. Res. Lett. 21 (2014), no. 3, 521-536. https://doi.org/10.4310/MRL.2014.v21.n3.a9 
  8. A. K. Lerner, S. Ombrosi, C. Perez, R. H. Torres, and R. Trujillo-Gonzalez, New maximal functions and multiple weights for the multilinear Calder'on-Zygmund theory, Adv. Math. 220 (2009), no. 4, 1222-1264. https://doi.org/10.1016/j.aim.2008.10.014 
  9. K. Li and W. Sun, Characterization of a two weight inequality for multilinear fractional maximal operators, Houston J. Math. 42 (2016), no. 3, 977-990. 
  10. K. Moen, Weighted inequalities for multilinear fractional integral operators, Collect. Math. 60 (2009), no. 2, 213-238. https://doi.org/10.1007/BF03191210 
  11. B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. https://doi.org/10.2307/1995882 
  12. B. Muckenhoupt and R. Wheeden, Weighted norm inequalities for fractional integrals, Trans. Amer. Math. Soc. 192 (1974), 261-274. https://doi.org/10.2307/1996833 
  13. G. Pan and L. Tang, New weighted norm inequalities for certain classes of multilinear operators and their iterated commutators, Potential Anal. 43 (2015), no. 3, 371-398. https://doi.org/10.1007/s11118-015-9477-2 
  14. Y. Pan and Q. Xue, Two weight characterizations for the multilinear local maximal operators, Acta Math. Sci. Ser. B (Engl. Ed.) 41 (2021), no. 2, 596-608. https://doi.org/10.1007/s10473-021-0219-9 
  15. C. Perez and E. Rela, A new quantitative two weight theorem for the Hardy-Littlewood maximal operator, Proc. Amer. Math. Soc. 143 (2015), no. 2, 641-655. https://doi.org/10.1090/S0002-9939-2014-12353-7 
  16. E. T. Sawyer, A characterization of a two-weight norm inequality for maximal operators, Studia Math. 75 (1982), no. 1, 1-11. https://doi.org/10.4064/sm-75-1-1-11 
  17. L. Tang, Weighted norm inequalities for pseudo-differential operators with smooth symbols and their commutators, J. Funct. Anal. 262 (2012), no. 4, 1603-1629. https://doi.org/10.1016/j.jfa.2011.11.016