Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

BOUNDEDNESS OF CALDERÓN-ZYGMUND OPERATORS ON INHOMOGENEOUS PRODUCT LIPSCHITZ SPACES

  • He, Shaoyong (Department of Mathematics Huzhou University) ;
  • Zheng, Taotao (Department of Mathematics Zhejiang University of Science and Technology)
  • Received : 2021.02.17
  • Accepted : 2022.01.06
  • Published : 2022.05.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we study the boundedness of a class of inhomogeneous Journé's product singular integral operators on the inhomogeneous product Lipschitz spaces. The consideration of such inhomogeneous Journé's product singular integral operators is motivated by the study of the multi-parameter pseudo-differential operators. The key idea used here is to develop the Littlewood-Paley theory for the inhomogeneous product spaces which includes the characterization of a special inhomogeneous product Besov space and a density argument for the inhomogeneous product Lipschitz spaces in the weak sense.

Keywords

Acknowledgement

The first author was supported by Zhejiang Provincial Natural Science Foundation of China (Grant No. LQ22A010018) and the second author was supported by National Natural Science Foundation of China (Grant Nos. 11626213, 11771399).

References

  1. S.-Y. A. Chang and R. Fefferman, A continuous version of duality of H1 with BMO on the bidisc, Ann. of Math. (2) 112 (1980), no. 1, 179-201. https://doi.org/10.2307/1971324
  2. S.-Y. A. Chang and R. Fefferman, The Calderon-Zygmund decomposition on product domains, Amer. J. Math. 104 (1982), no. 3, 455-468. https://doi.org/10.2307/2374150
  3. S.-Y. A. Chang and R. Fefferman, Some recent developments in Fourier analysis and Hp-theory on product domains, Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 1-43. https://doi.org/10.1090/S0273-0979-1985-15291-7
  4. J. Cheeger, Differentiability of Lipschitz functions on metric measure spaces, Geom. Funct. Anal. 9 (1999), no. 3, 428-517. https://doi.org/10.1007/s000390050094
  5. J. Chen, W. Ding, and G. Lu, Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces, Forum Math. 32 (2020), no. 4, 919-936. https://doi.org/10.1515/forum-2019-0319
  6. G. David and J.-L. Journe, A boundedness criterion for generalized Calderon-Zygmund operators, Ann. of Math. (2) 120 (1984), no. 2, 371-397. https://doi.org/10.2307/2006946
  7. W. Ding and G. Lu, Boundedness of inhomogeneous Journe's type operators on multi-parameter local Hardy spaces, Nonlinear Anal. 197 (2020), 111816, 31 pp. https://doi.org/10.1016/j.na.2020.111816
  8. W. Ding, G. Lu, and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 (2019), 352-380. https://doi.org/10.1016/j.na.2019.02.014
  9. W. Ding, G. Lu, and Y. Zhu, Discrete Littlewood-Paley-Stein characterization of multi-parameter local Hardy spaces, Forum Math. 31 (2019), no. 6, 1467-1488. https://doi.org/10.1515/forum-2019-0038
  10. R. Fefferman, Singular integrals on product Hp spaces, Rev. Mat. Iberoamericana 1 (1985), no. 2, 25-31. https://doi.org/10.4171/RMI/9
  11. R. Fefferman, Calderon-Zygmund theory for product domains: Hp spaces, Proc. Nat. Acad. Sci. U.S.A. 83 (1986), no. 4, 840-843. https://doi.org/10.1073/pnas.83.4.840
  12. R. Fefferman, Harmonic analysis on product spaces, Ann. of Math. (2) 126 (1987), no. 1, 109-130. https://doi.org/10.2307/1971346
  13. R. Fefferman and E. M. Stein, Singular integrals on product spaces, Adv. in Math. 45 (1982), no. 2, 117-143. https://doi.org/10.1016/S0001-8708(82)80001-7
  14. D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 (1979), no. 1, 27-42. http://projecteuclid.org/euclid.dmj/1077313253 https://doi.org/10.1215/S0012-7094-79-04603-9
  15. R. F. Gundy and E. M. Stein, Hp theory for the poly-disc, Proc. Nat. Acad. Sci. U.S.A. 76 (1979), no. 3, 1026-1029. https://doi.org/10.1073/pnas.76.3.1026
  16. Y. Han and Y. Han, Boundedness of composition operators associated with mixed homogeneities on Lipschitz spaces, Math. Res. Lett. 23 (2016), no. 5, 1387-1403. https://doi.org/10.4310/MRL.2016.v23.n5.a7
  17. Y. Han, Y. Han, J. Li, and C. Tan, Marcinkiewicz multipliers and Lipschitz spaces on Heisenberg groups, Canad. J. Math. 71 (2019), no. 3, 607-627. https://doi.org/10.4153/cjm-2018-003-0
  18. Y. Han, M. Lee, C. Lin, and Y. Lin, Calderon-Zygmund operators on product Hardy spaces, J. Funct. Anal. 258 (2010), no. 8, 2834-2861. https://doi.org/10.1016/j.jfa.2009.10.022
  19. Y. Han, G. Lu, and Z. Ruan, Boundedness criterion of Journe's class of singular integrals on multiparameter Hardy spaces, J. Funct. Anal. 264 (2013), no. 5, 1238-1268. https://doi.org/10.1016/j.jfa.2012.12.006
  20. Y. Han, G. Lu, and E. Sawyer, Flag Hardy spaces and Marcinkiewicz multipliers on the Heisenberg group, Anal. PDE 7 (2014), no. 7, 1465-1534. https://doi.org/10.2140/apde.2014.7.1465
  21. E. Harboure, O. Salinas, and B. Viviani, Boundedness of the fractional integral on weighted Lebesgue and Lipschitz spaces, Trans. Amer. Math. Soc. 349 (1997), no. 1, 235-255. https://doi.org/10.1090/S0002-9947-97-01644-9
  22. S. He and J. Chen, Three-parameter Hardy spaces associated with a sum of two flag singular integrals, Banach J. Math. Anal. 14 (2020), no. 4, 1363-1386. https://doi.org/10.1007/s43037-020-00067-w
  23. S. He and J. Chen, Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces, J. Pseudo-Differ. Oper. Appl. 11 (2020), no. 4, 1665-1683. https://doi.org/10.1007/s11868-020-00362-y
  24. S. He and J. Chen, Inhomogeneous Lipschitz spaces associated with flag singular integrals and their applications, Math. Inequal. Appl. 24 (2021), no. 4, 965-987. https://doi.org/10.7153/mia-2021-24-67
  25. S. He and J. Chen, Calder'on-Zygmund operators on multiparameter Lipschitz spaces of homogeneous type, Forum Math. 34 (2022), no. 1, 175-196. https://doi.org/10.1515/forum-2021-0204
  26. S. He and J. Chen, Mixed Lipschitz spaces and their applications, Acta Math. Sci. Ser. B (Engl. Ed.) 42 (2022), no. 2, 690-714. https://doi.org/10.1007/s10473-022-0217-6
  27. L. Hormander, Pseudo-differential operators, Comm. Pure Appl. Math. 18 (1965), 501-517. https://doi.org/10.1002/cpa.3160180307
  28. J.-L. Journe, Calderon-Zygmund operators on product spaces, Rev. Mat. Iberoamericana 1 (1985), no. 3, 55-91. https://doi.org/10.4171/RMI/15
  29. D. Muller, F. Ricci, and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. I, Invent. Math. 119 (1995), no. 2, 199-233. https://doi.org/10.1007/BF01245180
  30. D. Muller, F. Ricci, and E. M. Stein, Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups. II, Math. Z. 221 (1996), no. 2, 267-291. https://doi.org/10.1007/BF02622116
  31. A. Nagel, F. Ricci, and E. M. Stein, Singular integrals with flag kernels and analysis on quadratic CR manifolds, J. Funct. Anal. 181 (2001), no. 1, 29-118. https://doi.org/10.1006/jfan.2000.3714
  32. A. Nagel, F. Ricci, E. Stein, and S. Wainger, Singular integrals with flag kernels on homogeneous groups, I, Rev. Mat. Iberoam. 28 (2012), no. 3, 631-722. https://doi.org/10.4171/RMI/688
  33. A. Nagel, F. Ricci, E. M. Stein, and S. Wainger, Algebras of singular integral operators with kernels controlled by multiple norms, Mem. Amer. Math. Soc. 256 (2018), no. 1230, vii+141 pp. https://doi.org/10.1090/memo/1230
  34. J. Pipher, Journe's covering lemma and its extension to higher dimensions, Duke Math. J. 53 (1986), no. 3, 683-690. https://doi.org/10.1215/S0012-7094-86-05337-8
  35. V. S. Rychkov, Littlewood-Paley theory and function spaces with Alocp weights, Math. Nachr. 224 (2001), 145-180. https://doi.org/10.1002/1522-2616(200104)224:1<145::aid-mana145>3.0.co;2-2
  36. E. M. Stein, Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals, Princeton Mathematical Series, 43, Princeton University Press, Princeton, NJ, 1993.
  37. B. Street, Multi-parameter singular integrals, Annals of Mathematics Studies, 189, Princeton University Press, Princeton, NJ, 2014. https://doi.org/10.1515/9781400852758
  38. T. Zheng, J. Chen, J. Dai, S. He, and X. Tao, Calderon-Zygmund operators on homogeneous product Lipschitz spaces, J. Geom. Anal. 31 (2021), no. 2, 2033-2057. https://doi.org/10.1007/s12220-019-00331-y
  39. T. Zheng and X. Tao, T b criteria for Calderon-Zygmund operators on Lipschitz spaces with para-accretive functions, Publ. Math. Debrecen 95 (2019), no. 3-4, 487-503. https://doi.org/10.5486/pmd.2019.8582