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Some Weighted Estimates on Gaussian Measure Spaces

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Abstract

In this paper, we obtain the weighted boundedness for the local multi(sub)linear Hardy–Littlewood maximal operators and local multilinear fractional integral operators associated with the local Muckenhoupt weights on Gaussian measure spaces. We deal with these problems by introducing a new pointwise equivalent “radial” definitions of these local operators. Moreover, using a similar approach, we also get the weighted boundedness for the local fractional maximal operators with rough kernel and local fractional integral operators with rough kernel on Gaussian measure spaces.

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References

  1. Aimar, H., Forzani, L., Scotto, R.: On Riesz transforms and maximal functions in the context of Gaussian harmonic analysis. Trans. Am. Math. Soc. 359(5), 2137–2154 (2007)

    Article  MathSciNet  Google Scholar 

  2. Coifman, R.R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83(4), 569–645 (1977)

    Article  MathSciNet  Google Scholar 

  3. Cruz-Uribe, D., Martell, J.M., Pérez, C.: Sharp two-weight inequalities for singular integrals, with applications to the Hilbert transform and the Sarason conjecture. Adv. Math. 216(2), 647–676 (2007)

    Article  MathSciNet  Google Scholar 

  4. Ding, Y., Lu, S.: Weighted norm inequalities for fractional integral operators with rough kernel. Can. J. Math. 50(1), 29–39 (1998)

    Article  MathSciNet  Google Scholar 

  5. Duoandikoetxea, J.: Weighted norm inequalities for homogeneous singular integrals. Trans. Am. Math. Soc. 336(2), 869–880 (1993)

    Article  MathSciNet  Google Scholar 

  6. García-Cuerva, J., Martell, J.M.: Two-weight norm inequalities for maximal operators and fractional integrals on non-homogeneous spaces. Indiana Univ. Math. J. 50(3), 1241–1280 (2001)

    Article  MathSciNet  Google Scholar 

  7. García-Cuerva, J., de Francia, R., José, L.: Weighted norm inequalities and related topics. North-Holland Mathematics Studies, 116. Notas de Matemática [Mathematical Notes], 104. North-Holland Publishing Co., Amsterdam (1985). x+604 pp. ISBN: 0-444-87804-1

  8. Grafakos, L.: On multilinear fractional integrals. Stud. Math. 102(1), 49–56 (1992)

    Article  MathSciNet  Google Scholar 

  9. Grafakos, L., Kalton, N.: Some remarks on multilinear maps and interpolation. Math. Ann. 319(1), 151–180 (2001)

    Article  MathSciNet  Google Scholar 

  10. Grafakos, L.: Classical Fourier Analysis, 3rd edn. Graduate Texts in Mathematics, vol. 249. Springer, New York (2014). xviii+638 pp. ISBN: 978-1-4939-1193-6; 978-1-4939-1194-3

  11. Iida, T., Komori-Furuya, Y., Sato, E.: A note on multilinear fractional integrals. Anal. Theory Appl. 26(4), 301–307 (2010)

    Article  MathSciNet  Google Scholar 

  12. Kenig, C.E., Stein, E.M.: Multilinear estimates and fractional integration. Math. Res. Lett. 6(1), 1–15 (1999)

    Article  MathSciNet  Google Scholar 

  13. Kurtz, D.S., Wheeden, R.L.: Results on weighted norm inequalities for multipliers. Trans. Am. Math. Soc. 255, 343–362 (1979)

    Article  MathSciNet  Google Scholar 

  14. Lerner, A.K., Ombrosi, S., Pérez, C., Torres, R.H., Trujillo-González, R.: New maximal functions and multiple weights for the multilinear Calderón–Zygmund theory. Adv. Math. 220(4), 1222–1264 (2009)

    Article  MathSciNet  Google Scholar 

  15. Lin, C.-C., Stempak, K., Wang, Y.-S.: Local maximal operators on measure metric spaces. Publ. Mat. 57(1), 239–264 (2013)

    Article  MathSciNet  Google Scholar 

  16. Lin, H., Mao, S.: Weighted norm inequalities for local fractional integrals on Gaussian measure spaces. Potential Anal. 53(4), 1377–1401 (2020)

    Article  MathSciNet  Google Scholar 

  17. Liu, L., Yang, D.: BLO spaces associated with the Ornstein–Uhlenbeck operator. Bull. Sci. Math. 132(8), 633–649 (2008)

    Article  MathSciNet  Google Scholar 

  18. Liu, L., Yang, D.: Boundedness of Littlewood–Paley operators associated with Gauss measures. J. Inequal. Appl. Art. ID 643948, 41 pp (2010)

  19. Liu, L., Yang, D.: Characterizations of BMO associated with Gauss measures via commutators of local fractional integrals. Israel J. Math. 180, 285–315 (2010)

    Article  MathSciNet  Google Scholar 

  20. Liu, L., Yang, D.: Pointwise multipliers for Campanato spaces on Gauss measure spaces. Nagoya Math. J. 214, 169–193 (2014)

    Article  MathSciNet  Google Scholar 

  21. Liu, L., Sawano, Y., Yang, D.: Morrey-type spaces on Gauss measure spaces and boundedness for singular integrals. J. Geom. Anal. 24(2), 1007–1051 (2014)

    Article  MathSciNet  Google Scholar 

  22. Liu, L., Xiao, J., Yang, D., Yuan, W.: Gaussian Capacity Analysis. Lecture Notes in Mathematics, vol. 2225. Springer, Cham (2018)

    Book  Google Scholar 

  23. Lu, S., Ding, Y., Yan, D.: Singular integrals and related topics. World Scientific Publishing Co. Pte. Ltd., Hackensack (2007). viii+272 pp. ISBN: 978-981-270-623-2; 981-270-623-2

  24. Mauceri, G., Meda, S.: BMO and H1 for the Ornstein–Uhlenbeck operator. J. Funct. Anal. 252(1), 278–313 (2007)

    Article  MathSciNet  Google Scholar 

  25. Moen, K.: Weighted inequalities for multilinear fractional integral operators. Collect. Math. 60(2), 213–238 (2009)

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  27. Muckenhoupt, B., Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Am. Math. Soc. 192, 261–274 (1974)

    Article  MathSciNet  Google Scholar 

  28. Pérez, C.: Two weighted norm inequalities for Riesz potentials and uniform Lp-weighted Sobolev inequalities. Indiana Univ. Math. J. 39(1), 31–44 (1990)

    Article  MathSciNet  Google Scholar 

  29. Pérez, C.: Two weighted inequalities for potential and fractional type maximal operators. Indiana Univ. Math. J. 43(2), 663–683 (1994)

    Article  MathSciNet  Google Scholar 

  30. Sawyer, E., Wheeden, R.L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces. Am. J. Math. 114(4), 813–874 (1992)

    Article  MathSciNet  Google Scholar 

  31. Sawyer, E.T.: A characterization of a two-weight norm inequality for maximal operators. Stud. Math. 75(1), 1–11 (1982)

    Article  MathSciNet  Google Scholar 

  32. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton Mathematical Series, vol. 30. Princeton University Press, Princeton (1970)

    Google Scholar 

  33. Urbina-Romero, W.: Gaussian harmonic analysis. With a Foreword by Sundaram Thangavelu. Springer Monographs in Mathematics. Springer, Cham (2019). xix+477 pp. ISBN: 978-3-030-05596-7; 978-3-030-05597-4

  34. Wang, S., Zhou, H., Peng, L.: Local Muckenhoupt weights on Gaussian measure spaces. J. Math. Anal. Appl. 438(2), 790–806 (2016)

    Article  MathSciNet  Google Scholar 

  35. Watson, D.K.: Weighted estimates for singular integrals via Fourier transform estimates. Duke Math. J. 60(2), 389–399 (1990)

    Article  MathSciNet  Google Scholar 

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 11871452 and 12071473), Beijing Information Science and Technology University Foundation (Grant No. 2025031).

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Authors and Affiliations

  1. School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing, 100049, People’s Republic of China

    Boning Di & Dunyan Yan

  2. School of Applied Science, Beijing Information Science and Technology University, Beijing, 100192, People’s Republic of China

    Qianjun He

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  1. Boning Di
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  2. Qianjun He
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  3. Dunyan Yan
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Correspondence to Qianjun He.

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Communicated by Rosihan M. Ali.

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Di, B., He, Q. & Yan, D. Some Weighted Estimates on Gaussian Measure Spaces. Bull. Malays. Math. Sci. Soc. 44, 3907–3927 (2021). https://doi.org/10.1007/s40840-021-01153-4

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  • DOI: https://doi.org/10.1007/s40840-021-01153-4

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