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\(H^p\) Boundedness of Multilinear Spectral Multipliers on Stratified Groups

  • Jingxuan Fang
  • Jiman Zhao
Article
  • 9 Downloads

Abstract

As far as we know, there is no study about \(H^p\) boundedness of multilinear spectral multipliers on nilpotent Lie groups. In this paper, on stratified groups G, we prove a Hörmander type multiplier theorem for multilinear spectral multipliers on Hardy spaces, i.e., the boundedness from \(H^{p_1}\times H^{p_2} \times \cdots \times H^{p_N}\) to \(L^p\) with \(0<p_1,\ldots ,p_N,p \leqslant \infty \).

Keywords

Multilinear spectral multipliers Stratified groups Hörmander type condition 

Mathematics Subject Classification

42B15 43A80 47B40 

Notes

Acknowledgements

The authors would like to express great gratitude to the referees for the valuable comments and helpful suggestions.

References

  1. 1.
    Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Pure and Applied Mathematics, 2nd edn. Academic Press, New York (2003)Google Scholar
  2. 2.
    Alberto, R., Casas, S., Stein, E.M.: Marcinkiewicz multipliers in products of Heisenberg groups. Dissertations and theses—gradworks (2010)Google Scholar
  3. 3.
    Alexopoulos, G.: Spectral multipliers on Lie groups of polynomial growth. Proc. Am. Math. Soc. 120, 973–979 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Auscher, P., Carro, M.J.: Transference for radial multipliers and dimension free estimates. Trans. Am. Math. Soc. 342(2), 575–593 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Berg, J., Lofstrom, J.: Interpolation Spaces—An introduction. Springer, Berlin (1976)CrossRefzbMATHGoogle Scholar
  6. 6.
    Bernicot, F.: Uniform estimates for paraproducts and related multilinear multipliers. Rev. Mat. Iberoam. 25(3), 1055–1088 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Bernicot, F., Kova, V.: Sobolev norm estimates for a class of bilinear multipliers. Commun. Pure Appl. Anal. 13(3), 1305–1315 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Besov, O.V.: On Hörmander’s theorem on Fourier multipliers. Trudy Mat. Inst. Steklov. 173, 3–13 (1986)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chen, P., Duong, X.T., Yan, L.X.: \(L^p\)-bounds for Stein’s square functions associated to operators and applications to spectral multipliers. J. Math. Soc. Jpn. 65(2), 389–409 (2013)CrossRefzbMATHGoogle Scholar
  10. 10.
    Chen, L., Lu, G.Z., Luo, X.: Boundedness of multi-parameter Fourier multiplier operators on Triebel-Lizorkin and Besov-Lipschitz spaces. Nonlinear Anal. 134, 55–69 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Chen, P., Duong, X.T., Li, J., Ward, L.A., Yan, L.X.: Marcinkiewicz-type spectral multipliers on Hardy and Lebesgue spaces on product spaces of homogeneous type. J. Fourier Anal. Appl. 23(1), 21–64 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Christ, M.: \(L^p\) bounds for spectral multipliers on nilpotent groups. Trans. Am. Math. Soc. 328(1), 73–81 (1991)zbMATHGoogle Scholar
  13. 13.
    Christ, M., Müller, D.: On \(L^p\) spectral multipliers for a solvable lie group. Geom. Funct. Anal. 6(5), 860–876 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Christ, M., Grafakos, L., Honzík, P., Seeger, A.: Maximal functions associated with Fourier multipliers of Mikhlin-Hörmander type. Math. Z. 249(1), 223–240 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Cobos, F., Peetre, J., Persson, L.E.: On the connection between real and complex interpolation of quasi-banach spaces. Bull. Sci. Math. 122(1), 17–37 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Coifman, R.R., Meyer, Y.: On commutators of singular integrals and bilinear singular integrals. Trans. Am. Math. Soc. 212, 315–331 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Coifman, R.R., Meyer, Y.: Au delá des opérateurs pseudo-différentiels. Astérisque 57, 1–185 (1978)zbMATHGoogle Scholar
  18. 18.
    Coifman, R.R., Meyer, Y.: Commutateurs d’intégrales singuliéres et opérateurs multilinéaires. Ann. Inst. Fourier (Grenoble) 28, 177–202 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Courant, R., Hilbert, D.: Methods of mathematical physics, vol I. Phys. Today 7(5), 17–17 (1954)CrossRefGoogle Scholar
  20. 20.
    Duong, X.T.: From the \(L^1\) norms of the complex heat kernels to a Hörmander multiplier theorem for sub-Laplacians on nilpotent Lie groups. Pac. J. Math. 173(2), 413–424 (1996)CrossRefzbMATHGoogle Scholar
  21. 21.
    Fang, J.X., Zhao, J.M.: Multilinear and multiparameter spectral multipliers on stratified groups. Math. Methods Appl. 41(13), 5327–5344 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  22. 22.
    Folland, G.B., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Mathematical Notes, vol. 28. Princeton University Press, Princeton (1982)zbMATHGoogle Scholar
  23. 23.
    Furioli, G., Melzi, C., Veneruso, A.: Littlewood-Paley decompositions and Besov spaces on Lie groups of polynomial growth. Math. Nachr. 279(9–10), 1028–1040 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Gong, R.M., Yan, L.X.: Littlewood-Paley and spectral multipliers on weighted \(L^p\) spaces. J. Geom. Anal. 24(2), 873–900 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Grafakos, L.: Modern Fourier Analysis. Graduate Texts in Mathematics, vol. 250. Springer, New York (2008)Google Scholar
  26. 26.
    Grafakos, L., Mastylo, M.: Analytic families of multilinear operators. Nonlinear Anal. 107, 47–62 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Grafakos, L., Nguyen, H.V.: Multilinear Fourier multipliers with minimal Sobolev regularity, I. Colloq. Math. 144(1), 1–30 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Grafakos, L., Miyachi, A., Nguyen, H.V., Tomita, N.: Multilinear Fourier multipliers with minimal Sobolev regularity, II. J. Math. Soc. Jpn. 69(2), 529–562 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Grafakos, L., Miyachi, A., Tomita, N.: On multilinear Fourier multipliers of limited smoothness. Can. J. Math. 65(2), 299–330 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Grafakos, L., He, D. Q., Nguyen, H. V., Yan, L. X.: Multilinear multiplier theorems and applications. J. Fourier. Anal. Appl. (2018).  https://doi.org/10.1007/s00041-018-9606-6
  31. 31.
    Hilbert, D.: Grundzüge einer allgemeinen Theorie der linearen Integralgleichungen. (German) Chelsea Publishing Company, New York, NY, pp xxvi+282 (1953)Google Scholar
  32. 32.
    Hömander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Kolomoitsev, Y.S.: Generalization of one sufficient condition for Fourier multipliers. Ukrainian Math. J. 64(10), 1562–1571 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Kolomoitsev, Y.S.: Multiplicative sufficient conditions for Fourier multipliers. Izv. Math. 78(2), 354–374 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Lin, C.C.: \(L^p\) multipliers and their \(H^1-L^1\) estimates on the Heisenberg group. Rev. Mat. Iberoam. 11(2), 269–308 (1995)CrossRefGoogle Scholar
  36. 36.
    Lu, S.Z., Yang, D.C., Zhou, Z.S.: Some multiplier theorems for non-isotropic \(H^p({ {R}}^n)\). J. Beijing Norm. Univ. 33(1), 1–9 (1997)Google Scholar
  37. 37.
    Marcinkiewicz, J.: Sur les multiplicateurs des séries de Fourier. Studia Math. 8, 78–91 (1939)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Martini, A.: Analysis of joint spectral multipliers on Lie groups of polynomial growth. Ann. Inst. Fourier 62(4), 1215–1263 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Martini, A.: Algebras of differential operators on Lie groups and spectral multipliers. PhD Thesis, 2010Google Scholar
  40. 40.
    Mauceri, G., Meda, S.: Vector-valued multipliers on stratified groups. Rev. Mat. Iberoamericana 6, 141–154 (1990)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Michele, L.D., Mauceri, G.: \(H^p\) multipliers on stratified groups. Ann. Mat. Pura Appl. 148(4), 353–366 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  42. 42.
    Mihlin, S.G.: On the multipliers of Fourier integrals. Dokl. Akad. Nauk SSSR 1956(109), 701–703 (1956)MathSciNetGoogle Scholar
  43. 43.
    Miyachi, A., Tomita, N.: Minimal smoothness conditions for bilinear Fourier multipliers. Rev. Mat. Iberoamer. 29, 495–530 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  44. 44.
    Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and two-parameter structures on Heisenberg (-type) groups, I. Invent. Math. 119(1), 199–233 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  45. 45.
    Noi, T.: Fourier multiplier theorems for Besov and Triebel-Lizorkin spaces with variable exponents. Math. Inequal. Appl. 17(1), 49–74 (2014)MathSciNetzbMATHGoogle Scholar
  46. 46.
    Pini, R.: A multiplier theorem for H-type groups. Studia Math. 100(1), 39–49 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Song, N.Q., Liu, H.P., Zhao, J.M.: Bilinear spectral multipliers on Heisenberg groups (preprint)Google Scholar
  48. 48.
    Stein, E.M.: Spectral multipliers and multiple-parameter structures on the Heisenberg group. Journées équations aux dérivées partielles 1995, 1–15 (1995)CrossRefzbMATHGoogle Scholar
  49. 49.
    Tartar, L.: An Introduction to Sobolev Spaces and Interpolation Spaces. Lecture Notes of the Unione Matematica Italiana, vol. 3, pp. 14–21. Springer, Berlin (2007)Google Scholar
  50. 50.
    Tomita, N.: Hörmander type multiplier theorem for multilinear operators. J. Funct. Anal. 259(8), 2028–2044 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  51. 51.
    Tréves, F.: Topological Vector Spaces, Distributions and Kernels. Academic Press, New York (1967)zbMATHGoogle Scholar
  52. 52.
    Wendel, J.G.: Left centralizers and isomorphisms of group algebras. Pac. J. Math. 2(2), 251–261 (1952)MathSciNetCrossRefzbMATHGoogle Scholar
  53. 53.
    Wróbel, B.: Joint spectral multipliers for mixed systems of operators. J. Fourier Anal. Appl. 23(2017), 245–287 (2015)MathSciNetzbMATHGoogle Scholar
  54. 54.
    Yabuta, K.: Multilinear Littlewood-Paley operators and multilinear Fourier multipliers. S\({\bar{\rm u}}\)rikaisekikenky\({\bar{\rm u}}\)sho K\({\bar{\rm o}}\)ky\({\bar{\rm u}}\)roku 1235, 54–60 (2001)Google Scholar
  55. 55.
    Yang, D. C., Yuan, W., Zhuo, C. Q.: Fourier multipliers on Triebel–Lizorkin–type spaces. J. Funct. Spaces Appl. Art. ID 431016, 37 (2012).  https://doi.org/10.1155/2012/431016
  56. 56.
    Zhao, G.P., Chen, J.C., Fan, D.S., Guo, W.C.: Unimodular Fourier multipliers on homogeneous Besov spaces. J. Math. Anal. Appl. 425(1), 536–547 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  57. 57.
    Zhao, G.P., Chen, J.C., Fan, D.S., Guo, W.C.: Sharp estimates of unimodular multipliers on frequency decomposition spaces. Nonlinear Anal. Theor. 142, 26–47 (2016)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2019

Authors and Affiliations

  1. 1.Key Laboratory of Mathematics and Complex Systems, Ministry of Education Institution of Mathematics and Mathematical Education, School of Mathematical SciencesBeijing Normal UniversityBeijingChina

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