Skip to main content
SpringerLink
Log in

Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces

  • Published:
Journal of Pseudo-Differential Operators and Applications Aims and scope Submit manuscript

Abstract

The main purpose of this paper is to introduce the multi-parameter Lipschitz spaces and characterize it via the Littlewood–Paley theory. As an application to the multi-parameter Lipschitz spaces, we derive the boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Calderón, A.P., Vaillancourt, R.: A class of bounded pseudo-differential operators. Proc. Nat. Acad. Sci. U.S.A. 69, 1185–1187 (1972)

    MathSciNet  MATH  Google Scholar 

  2. Chang, S.Y., Fefferman, R.: A continuous version of duality of \(H^1\) with BMO on the bidisc. Ann. Math. 112(1), 179–201 (1980)

    MathSciNet  MATH  Google Scholar 

  3. Chang, S.Y., Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Am. J. Math. 104(3), 455–468 (1982)

    MATH  Google Scholar 

  4. Chang, S.Y., Fefferman, R.: Some recent developments in fourier analysis and \(H^p\) theory on product domains. Bull. Am. Math. Soc. 12(1), 1–43 (1985)

    MATH  Google Scholar 

  5. Chen, J., Ding, W., Lu, G.: Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces. Forum Math. 32(4), 919–936 (2020)

  6. Dai, W., Lu, G.: \(L^p\) estimates for multi-linear and multi-parameter pseudo-differential operators. Bull. Math. Sci. France 143(3), 567–597 (2015)

    MATH  Google Scholar 

  7. David, G., Journé, J.L.: A boundedness criterion for generalized Calderón-Zygmund operators. Ann. Math. 120(2), 371–397 (1984)

    MathSciNet  MATH  Google Scholar 

  8. Ding, W., Lu, G., Zhu, Y.: Discrete Littlewood-Paley-Stein characterization of multi-parameter local Hardy spaces. Forum Math. 31(6), 1467–1488 (2019)

    MathSciNet  MATH  Google Scholar 

  9. Ding, W., Lu, G., Zhu, Y.: Multi-parameter local Hardy spaces. Nonlinear Anal. 184, 352–380 (2019)

    MathSciNet  MATH  Google Scholar 

  10. Èskin, G.I.: Degenerate elliptic pseudodifferential equations of principal type. Mat. Sb. (N.S.) 82(124), 585–628 (1970)

  11. Fefferman, C.: \(L^p\) bounds for pseudo-differentialoperators. Israel J. Math. 14, 413–417 (1973)

    MathSciNet  MATH  Google Scholar 

  12. Fefferman, R.: Harmonic analysis on product spaces. Ann. Math. 126(1), 109–130 (1987)

    MathSciNet  MATH  Google Scholar 

  13. Fefferman, R., Pipher, J.: Multiparameter operators and sharp weighted inequalities. Am. J. Math. 119(2), 337–369 (1997)

    MathSciNet  MATH  Google Scholar 

  14. Fefferman, R., Stein, E.M.: Singular integrals on product spaces. Adv. Math. 45(2), 117–143 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  15. Goldberg, D.: A local version of real Hardy spaces. Duke Math. J. 46(1), 27–42 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  16. Gundy, R.F., Stein, E.M.: \(H^p\) theory for the poly-disk. Proc. Nat. Acad. Sci. 76(3), 1026–1029 (1979)

    MathSciNet  MATH  Google Scholar 

  17. Han, Y., Han, Y.: Boundedness of composition operators associated with mixed homogeneities on Lipschitz spaces. Math. Res. Lett. 23(5), 1387–1403 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  18. Han, Y., Han, Y., Li, J., Tan, C.: Marcinkiewicz multipliers and Lipschitz Spaces on Heisenberg groups. Can. J. Math. 71(3), 607–627 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  19. Han, Y., Lee, M., Lin, C., Lin, Y.: Calderón-Zygmund operators on product Hardy spaces. J. Funct. Anal. 258(8), 2834–2861 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  20. Han, Y., Li, J., Lin, C., Tan, C.: Singular integrals associated with Zygmund dilations. J. Geom. Anal. 29(3), 2410–2455 (2019)

    MathSciNet  MATH  Google Scholar 

  21. Han, Y., Lu, G., Ruan, Z.: Boundedness criterion of Journes class of singular integrals on multiparameter Hardy spaces. J. Funct. Anal. 264(5), 1238–1268 (2013)

    MathSciNet  MATH  Google Scholar 

  22. Han, Y., Lu, G., Sawyer, E.: Flag Hardy space and Marcinkiewicz multipliers on the Heisnberg group. Anal. PDE 7(7), 1465–1534 (2014)

    MathSciNet  MATH  Google Scholar 

  23. He, S., Chen, J.: Three-parameter Hardy spaces associated with a sum of two flag singular integrals. Banach. J. Math. Anal. (2020). https://doi.org/10.1007/s43037-020-00067-w

  24. Hong, Q., Lu, G.: Symbolic calculus and boundedness of multi-parameter and multi-linear pseudo-differential operators. Adv. Nonlinear Stud. 14(4), 1055–1082 (2014)

    MathSciNet  MATH  Google Scholar 

  25. Hong, Q., Lu, G.: Weighted \(L^p\) estimates for rough bi-parameter Fourier integral operators. J. Differ. Equ. 265(3), 1097–1127 (2018)

    MATH  Google Scholar 

  26. Hong, Q., Lu, G., Zhang, L.: \(L^p\) boundedness of rough bi-parameter Fourier integral operators. Forum Math. 30(1), 87–107 (2018)

    MathSciNet  MATH  Google Scholar 

  27. Hörmander, L.: Pseudo-differential operators. Commun. Pure Appl. Math. 18, 501–517 (1965)

    MathSciNet  MATH  Google Scholar 

  28. Hörmander, L.: Pseudo-differential operators and hypoelliptic equations. Proc. Symp. Pure Math. 10, 138–183 (1967)

    MathSciNet  MATH  Google Scholar 

  29. Hörmander, L.: Fourier integral operators I. Acta Math. 127(1–2), 79–183 (1971)

    MathSciNet  MATH  Google Scholar 

  30. Journé, J.L.: Calderón-Zygmund operators on product spaces. Rev. Math. Iberoam. 1(3), 55–91 (1985)

    MATH  Google Scholar 

  31. Kato, T., Ponce, G.: Commutator estimates and the Euler and NavierStokes equations. Commun. Pure Appl. Math. 41(7), 891–907 (1988)

    MATH  Google Scholar 

  32. Kohn, J., Nirenberg, L.: An algebra of pseudo-differential operators. Commun. Pure Appl. Math. 18, 269–305 (1965)

    MathSciNet  MATH  Google Scholar 

  33. Lin, Y., Lu, S.: Pseduo-differential operators on Sobolev and Lipschitz spaces. Acta Math. Sin. (Engl. Ser.) 26(1), 131–142 (2010)

  34. Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter strucure on Heisenberg(-type) groups I. Invent. Math. 119(2), 119–233 (1995)

    Google Scholar 

  35. Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter strucure on Heisenberg(-type) groups II. Math. Z. 221(2), 267–291 (1996)

    MathSciNet  MATH  Google Scholar 

  36. Päivärinta, L., Somersalo, E.: A generalization of the Calderón-Vaillancourt theorem to \(L^p\) and \(h^p\). Math. Nachr. 138, 145–156 (1988)

    MathSciNet  MATH  Google Scholar 

  37. Nagel, A., Ricci, F., Stein, E.M.: Singular integrals with flag kernel and analysis on quadratic CR manifolds. J. Funct. Anal 181(1), 29–118 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nagel, A., Ricci, F., Stein, E.M.: Singular integrals with flag kernel on homogeneous group : I. Rev. Math. Iberoam. 28(3), 673–722 (2012)

    Article  MathSciNet  Google Scholar 

  39. Nagel, A., Ricci, F., Stein, E. M.,Wainger, S.: Algebrals of singular integrals operator with kernels controlled by multiple norms. Mem. Am. Math. Soc. 256(1230), vii+141 (2018)

  40. Nagel, A., Stein, E.M.: A new class of pseduo-differential operators. Proc. Nat. Acad. Sci. U.S.A. 75(2), 582–585 (1978)

    MathSciNet  MATH  Google Scholar 

  41. Pipher, J.: Journé’s covering lemma and its extension to higher dimensions. Duke Math. J. 53(3), 683–690 (1986)

    MathSciNet  MATH  Google Scholar 

  42. Ricci, F., Stein, E.M.: Multiparameter singular integrals and maximal functions. Ann. Inst. Fourier (Grenoble) 42(3), 637–670 (1992)

    MathSciNet  MATH  Google Scholar 

  43. Seeger, A., Sogge, C.D., Stein, E.M.: Regularity properties of Fourier integral operators. Ann. Math. (2) 134(2), 231–251 (1991)

  44. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality, and Oscillatory Integrals. Princeton University Press, Princeton (1993)

    MATH  Google Scholar 

  45. Xu, C., Huang, L.: Boundedness of bi-parameter pseudo-differential operaators on bi-parameter \(\alpha \)-modulation spaces. Nonlinear Anal. 180, 20–40 (2019)

    MathSciNet  MATH  Google Scholar 

  46. Zheng, T., Chen, J., Dai, J., He, S., Tao, X.: Calderón-Zygmund operators on homogeneous product Lipschitz spaces. J. Geom. Anal. (2019). https://doi.org/10.1007/s12220-019-00331-y

    Article  Google Scholar 

Download references

Acknowledgements

The authors would like to thank the referee for his/her very careful reading and helpful comments which have improved the paper sub-stantially.

Author information

Authors and Affiliations

  1. Department of Mathematics, Zhejiang Normal University, Jinhua, China

    Shaoyong He & Jiecheng Chen

Authors
  1. Shaoyong He
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Jiecheng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Shaoyong He.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This research was funded by National Natural Science Foundation of China (Grant No. 11671363).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

He, S., Chen, J. Boundedness of multi-parameter pseudo-differential operators on multi-parameter Lipschitz spaces. J. Pseudo-Differ. Oper. Appl. 11, 1665–1683 (2020). https://doi.org/10.1007/s11868-020-00362-y

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11868-020-00362-y

Keywords

Mathematics Subject Classification

Search

Navigation