Skip to main content
SpringerLink
Log in

Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type

  • Published:
Journal of Fourier Analysis and Applications Aims and scope Submit manuscript

Abstract

Let \(X_1\) and \(X_2\) be metric spaces equipped with doubling measures and let \(L_1\) and \(L_2\) be nonnegative self-adjoint operators acting on \(L^2(X_1)\) and \(L^2(X_2)\) respectively. We study multivariable spectral multipliers \(F(L_1, L_2)\) acting on the Cartesian product of \(X_1\) and \(X_2\). Under the assumptions of the finite propagation speed property and Plancherel or Stein–Tomas restriction type estimates on the operators \(L_1\) and \(L_2\), we show that if a function F satisfies a Marcinkiewicz-type differential condition then the spectral multiplier operator \(F(L_1, L_2)\) is bounded from appropriate Hardy spaces to Lebesgue spaces on the product space \(X_1\times X_2\). We apply our results to the analysis of second-order elliptic operators in the product setting, specifically Riesz-transform-like operators and double Bochner–Riesz means.

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. Auscher, P.: On Necessary and Sufficient Conditions for \(L^p\)-Estimates of Riesz Transforms Associated to Elliptic Operators on \({\mathbb{R}}\) and Related Estimates, vol. 186(871). Memoirs of the American Mathematical Society (2007)

  2. Auscher, P., McIntosh, A., Russ, E.: Hardy spaces of differential forms on Riemannian manifolds. J. Geom. Anal. 18, 192–248 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bernicot, F., Grafakos, L., Song, L., Yan, L.X.: The bilinear Bochner-Riesz problem. J. Anal. Math. 127, 179–217 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Bownik, M., Li, B.D., Yang, D.C., Zhou, Y.: Weighted anisotropic product Hardy spaces and boundedness of sublinear operators. Math. Nachr. 283(3), 392–442 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  5. Carbery, A., Seeger, A.: \(H^p\)- and \(L^p\)-variants of multiparameter Calderón-Zygmund theory. Trans. Am. Math. Soc. 334, 719–747 (1992)

    MATH  Google Scholar 

  6. Carleson, L.: A counterexample for measures bounded on \(H^p\) for the bi-disc. Mittag Leffler Report No. 7 (1974)

  7. Chang, D.C., Yang, D.C., Zhou, Y.: Boundedness of sublinear operators on product Hardy spaces and its application. J. Math. Soc. Jpn 62(1), 321–353 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  9. Chang, S.-Y.A., Fefferman, R.: The Calderón-Zygmund decomposition on product domains. Am. J. Math. 104, 445–468 (1982)

    Article  MATH  Google Scholar 

  10. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplacian and the geometry of complete Riemannian manifolds. J. Differ. Geom. 17, 15–53 (1982)

    MATH  Google Scholar 

  11. Chen, L.K.: The multiplier operators on the product spaces. Ill. J. Math. 38, 420–433 (1994)

    MathSciNet  MATH  Google Scholar 

  12. Chen, P., Duong, X.T., Li, J., Ward, L.A., Yan, L.: Product Hardy spaces associated to operators with heat kernel bounds on spaces of homogeneous type. Math. Z. (2015). doi:10.1007/s00209-015-1577-6

  13. Chen, P., Ouhabaz, E.M., Sikora, A., Yan, L.X.: Endpoint estimates for Bochner-Riesz means and sharp spectral multipliers. J. d’Analyse Mathématique (to appear)

  14. Christ, M.: A \(T(b)\) theorem with remarks on analytic capacity and the Cauchy integral. Colloq. Math. 61, 601–628 (1990)

    MathSciNet  MATH  Google Scholar 

  15. Christ, M.: \(L^p\) bounds for spectral multipliers on nilpotent groups. Trans. Am. Math. Soc. 328(1), 73–81 (1991)

    MathSciNet  MATH  Google Scholar 

  16. Coifman, R.R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Étude de certaines intégrals singulières. Lecture Notes in Mathematics, vol. 242. Springer, Berlin (1971)

    Book  MATH  Google Scholar 

  17. Coulhon, T., Sikora, A.: Gaussian heat kernel upper bounds via Phragmén-Lindelöf theorem. Proc. Lond. Math. Soc. 96, 507–544 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Deng, D.G., Song, L., Tan, C.Q., Yan, L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds on product domains. J. Geom. Anal. 17, 455–483 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  19. Duong, X.T., Li, J., Yan, L.X.: Endpoint estimates for singular integrals with non-smooth kernels on product spaces, preprint, arXiv:1509.07548 (2014)

  20. Duong, X.T., Ouhabaz, E.M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196, 443–485 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Duong, X.T., Yan, L.X.: Duality of Hardy and BMO spaces associated with operators with heat kernel bounds. J. Am. Math. Soc. 18, 943–973 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  22. Duong, X.T., Yan, L.X.: Spectral multipliers for Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates. J. Math. Soc. Jpn. 63, 295–319 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  23. Fefferman, R.: Calderón-Zygmund theory for product domains: \(H^p\) spaces. Proc. Natl. Acad. Sci. USA 83, 840–843 (1986)

    Article  MATH  Google Scholar 

  24. Folland, G., Stein, E.M.: Hardy Spaces on Homogeneous Groups. Princeton University Press, Princeton (1982)

    MATH  Google Scholar 

  25. Grafakos, L., Honzk, P., Seeger, A.: On maximal functions for Mikhlin-Hrmander multipliers. Adv. Math. 204(2), 363–378 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Grafakos, L., Si, Z.Y.: The Hörmander multiplier theorem for multilinear operators. J. Reine Angew. Math. 668, 133–147 (2012)

    MATH  Google Scholar 

  27. Guillarmou, C., Hassell, A., Sikora, A.: Restriction and spectral multiplier theorems on asymptotically conic manifolds. Anal. PDE 6, 893–950 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  28. Gundy, R., Stein, E.M.: \(H^p\) theory for the poly-disc. Proc. Natl. Acad. Sci. 76, 1026–1029 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  29. Han, Y.S., Li, J., Lin, C.C.: Criterions of the \(L^2\) boundedness and sharp endpoint estimates for singular integral operators on product spaces of homogeneous type. Ann. Scuola Norm. Sup. Pisa (2015). doi:10.2422/2036-2145.201411_002

  30. Hofmann, S., Lu, G.Z., Mitrea, D., Mitrea, M., Yan, L.X.: Hardy Spaces Associated to Non-negative Self-adjoint Operators Satisfying Davies-Gaffney Estimates, vol. 214(1007).Memoirs of the American Mathematical Society (2011)

  31. Hofmann, S., Mayboroda, S.: Hardy and BMO spaces associated to divergence form elliptic operators. Math. Ann. 344, 37–116 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  32. Hofmann, S., Mayboroda, S., McIntosh, A.: Second order elliptic operators with complex bounded measurable coefficients in \(L^p\), Sobolev and Hardy spaces. Ann. Sci. École Norm. Sup. 44, 723–800 (2011)

    MathSciNet  MATH  Google Scholar 

  33. Hörmander, L.: Estimates for translation invariant operators in \(L^p\) spaces. Acta Math. 104, 93–140 (1960)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  35. Journé, J.-L.: A covering lemma for product spaces. Proc. Am. Math. Soc. 96, 593–598 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kunstmann, P.C., Uhl, M.: Spectral multiplier theorems of Hörmander type on Hardy and Lebesgue spaces. J. Oper. Theory 73, 27–69 (2015)

    Article  MATH  Google Scholar 

  37. Liskevich, V., Sobol, Z., Vogt, H.: On the \(L^p\) theory of \(C^0\)-semigroups associated with second-order elliptic operators II. J. Funct. Anal. 193, 55–76 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  38. Martini, A.: Analysis of joint spectral multipliers on Lie groups of polynomial growth. Ann. Inst. Fourier (Grenoble) 62, 1215–1263 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  39. Müller, D., Ricci, F., Stein, E.M.: Marcinkiewicz multipliers and multi-parameter structure on Heisenberg (-type) groups, I. Invent. Math. 119, 199–233 (1995)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  41. Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs, vol. 31. Princeton University Press, Princeton (2004)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  43. Sikora, A.: Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247, 643–662 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  44. Sikora, A.: Multivariable spectral multipliers and analysis of quasielliptic operators on fractals. Indiana Univ. Math. J. 58, 317–334 (2009)

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  46. Stein, E.M.: Harmonic Analysis: Real Variable Methods, Orthogonality and Oscillatory Integrals, with the assistance of Timothy S. Murphy, Princeton Mathematical Series, Monographs in Harmonic Analysis, III, vol. 43. Princeton University Press, Princeton (1993)

  47. Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge University Press, London (1993)

    Book  MATH  Google Scholar 

  48. Yosida, K.: Functional Analysis, 5th edn. Springer, Berlin (1978)

    Book  MATH  Google Scholar 

Download references

Acknowledgments

P. Chen, X. T. Duong, J. Li and L. A. Ward are supported by the Australian Research Council (ARC) under Grant No. ARC-DP120100399. P. Chen is also supported by the NNSF of China, Grant No. 11501583. J. Li is also supported by the NNSF of China, Grant No. 11001275. L. X. Yan is supported by the NNSF of China, Grant No. 11371378. Part of this work was done during L. X. Yan’s stay at Macquarie University and visit to the University of South Australia. L. X. Yan would like to thank Macquarie University and the University of South Australia for their hospitality.

Author information

Authors and Affiliations

  1. Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

    Peng Chen & Lixin Yan

  2. Department of Mathematics, Macquarie University, North Ryde, NSW, 2109, Australia

    Xuan Thinh Duong & Ji Li

  3. School of Information Technology and Mathematical Sciences, University of South Australia, Mawson Lakes, SA, 5095, Australia

    Lesley A. Ward

Authors
  1. Peng Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xuan Thinh Duong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ji Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Lesley A. Ward
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Lixin Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Xuan Thinh Duong.

Additional information

Communicated by Dachun Yang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Chen, P., Duong, X.T., Li, J. et al. Marcinkiewicz-Type Spectral Multipliers on Hardy and Lebesgue Spaces on Product Spaces of Homogeneous Type. J Fourier Anal Appl 23, 21–64 (2017). https://doi.org/10.1007/s00041-016-9460-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-016-9460-3

Keywords

Mathematics Subject Classification

Search

Navigation