Skip to main content
SpringerLink
Log in

Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in LP

  • Published:
Journal d'Analyse Mathématique Aims and scope

Abstract

Perturbed Hodge-Dirac operators and their holomorphic functional calculi, as investigated in the papers by Axelsson, Keith and the second author, provided insight into the solution of the Kato square-root problem for elliptic operators in L2 spaces and allowed for an extension of these estimates to other systems with applications to non-smooth boundary value problems. In this paper, we determine conditions under which such operators satisfy conical square function estimates in a range of Lp spaces, thus allowing us to apply the theory of Hardy spaces associated with an operator to prove that they have a bounded holomorphic functional calculus in those Lp spaces. We also obtain functional calculus results for restrictions to certain subspaces, for a larger range of p. This provides a framework for obtaining Lp results on perturbed Hodge Laplacians, generalising known Riesz transform bounds for an elliptic operator L with bounded measurable coefficients, one Sobolev exponent below the Hodge exponent, and Lp bounds on the square-root of L by the gradient, two Sobolev exponents below the Hodge exponent. Our proof shows that the heart of the harmonic analysis in L2 extends to Lp for all p ∈ (1,∞), while the restrictions in p come from the operator-theoretic part of the L2 proof. In the course of our work, we obtain some results of independent interest about singular integral operators on tent spaces and about the relationship between conical and vertical square functions.

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. S. Ajiev, Extrapolation of the functional calculus of generalized Dirac operators and related embedding and Littlewood-Paley-type theorems, J. Aust. Math. Soc. 83 (2007), 297–326.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Albrecht, X. Duong, and A. McIntosh, Operator theory and harmonic analysis, in Instructional Workshop on Analysis and Geometry, Part III (Canberra, 1995), Proc. Centre Math. Appl. Austral. Nat. Univ. 34 (1996), 77–136.

    MATH  Google Scholar 

  3. A. Amenta, Interpolation and embeddings of weighted tent spaces, J. Fourier Anal. Appl. (2017), to appear. DOI 10.1007/s00041-017-9521-2.

    Google Scholar 

  4. P. Auscher, On necessary and sufficient conditions for L p-estimates of Riesz transforms associated to elliptic operators on Rn and related estimates, Mem. Amer. Math. Soc. 871 (2007).

  5. P. Auscher, Change of angle in tent spaces, C. R. Math. Acad. Sci. Paris 349 (2011), 297–301.

    Article  MathSciNet  MATH  Google Scholar 

  6. P. Auscher and A. Axelsson, Weighted maximal regularity estimates and solvability of nonsmooth elliptic systems I, Invent. Math. 184 (2011), 47–115.

    Article  MathSciNet  MATH  Google Scholar 

  7. P. Auscher, A. Axelsson, and S. Hofmann., Functional calculus of Dirac operators and complex perturbations of Neumann and Dirichlet problems, J. Funct. Anal. 255 (2008), 374–448.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Auscher, A. Axelsson, and A. McIntosh, Solvability of elliptic systems with square integrable boundary data, Ark. Mat. 48 (2010), 253–287.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Auscher, A. Axelsson, and A. McIntosh, On a quadratic estimate related to the Kato conjecture and boundary value problems, Harmonic Analysis and Partial Differential Equations, Amer. Math. Soc., Providence, RI, 2010, pp. 105–129.

    MATH  Google Scholar 

  10. P. Auscher, S. Hofmann, M. Lacey, A. McIntosh, and P. Tchamitchian, The solution of the Kato square root problem for second order elliptic operators on Rn, Ann. of Math. (2) 156 (2002), 633–654.

    Article  MathSciNet  MATH  Google Scholar 

  11. P. Auscher, S. Hofmann, and J. M. Martell. Vertical versus conical square functions, Trans. Amer. Math. Soc. 364 (2012), 5469–5489.

    Article  MathSciNet  MATH  Google Scholar 

  12. P. Auscher, C. Kriegler, S. Monniaux, and P. Portal, Singular integral operators on tent spaces, J. Evol. Equ. 12 (2012), 741–765.

    Article  MathSciNet  MATH  Google Scholar 

  13. P. Auscher and J. M. Martell, Weighted norm inequalities, off-diagonal estimates and elliptic operators Part II: Off-diagonal estimates on spaces of homogeneous type, J. Evol. Equ. 7 (2007) 265–316.

    Article  MathSciNet  MATH  Google Scholar 

  14. P. Auscher, A. McIntosh, and A. Morris, Calderón reproducing formulas and applications to Hardy spaces, Rev. Mat. Iberoam. 31 (2015), 865–900.

    Article  MathSciNet  MATH  Google Scholar 

  15. P. Auscher, A. McIntosh, and A. Nahmod, Holomorphic functional calculi of operators, quadratic estimates and interpolation, Indiana Univ. Math. J. 46 (1997), 375–403.

    MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  17. P. Auscher and S. Stahlhut, Remarks on functional calculus for perturbed first order Dirac operator, Oper. Theory Adv. Appl. 240, Birkhäuser/Springer Basel AG, 2014, pp. 31–43.

    Article  MathSciNet  MATH  Google Scholar 

  18. P. Auscher and S. Stahlhut. A priori estimates for boundary value elliptic problems via first order systems, arXiv preprint arXiv:1403.5367.

  19. A. Axelsson, S. Keith, and A. McIntosh, Quadratic estimates and functional calculi of perturbed Dirac operators, Invent. Math. 163 (2006), 455–497.

    Article  MathSciNet  MATH  Google Scholar 

  20. L. Bandara and A. McIntosh. The Kato square root problem on vector bundles with generalised bounded geometry, J. Geom. Anal. 26 (2016), 428–462.

    Article  MathSciNet  MATH  Google Scholar 

  21. S. Blunck and P. Kunstmann, Calderón-Zygmund theory for non-integral operators and the H∞ functional calculus, Rev. Mat. Iberoam. 19 (2003), 919–942.

    Article  MATH  Google Scholar 

  22. W. Cohn and I. Verbitsky, Factorization of tent spaces and Hankel operators, J. Funct. Anal. 175 (2000), 308–329.

    Article  MathSciNet  MATH  Google Scholar 

  23. R. Coifman, Y. Meyer, and E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 (1985), 304–335.

    Article  MathSciNet  MATH  Google Scholar 

  24. R. Coifman and G. Weiss, Analyse harmonique sur certains espaces homogènes, Springer-Verlag, 1971.

    Book  MATH  Google Scholar 

  25. M. Cowling, I. Doust, A. McIntosh, and A. Yagi, Banach space operators with a bounded H∞ functional calculus, J. Austral. Math. Soc. Ser. A 60 (1996), 51–89.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  27. C. Fefferman and E. Stein, H p spaces of several variables, Acta Math. 129 (1972) 137–193.

    Article  MathSciNet  MATH  Google Scholar 

  28. L. Grafakos, Classical Fourier Analysis, Third edition, Springer, New York, 2014.

    Book  MATH  Google Scholar 

  29. M. Haase, The Functional Calculus for Sectorial Operators, Birkhäuser Verlag, Basel, 2006.

    Book  MATH  Google Scholar 

  30. E. Harboure, J. L. Torrea, and B. E. Viviani, A vector-valued approach to tent spaces, J. Anal. Math. 56 (1991), 125–140.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. Hofmann, G. Lu, D. Mitrea, M. Mitrea, and L. Yan, Hardy spaces associated to non-negative self-adjoint operators satisfying Davies-Gaffney estimates, Mem. Amer. Math. Soc. 214 (2011).

  32. S. Hofmannm and J. M. Martell, Lp bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003) 497–515.

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  34. S. Hofmann, S. Mayboroda, and A. McIntosh. Second order elliptic operators with complex bounded measurable coefficients in L p, Sobolev and Hardy spaces, Ann. Sci. Éc. Norm. Supér. (4) 44 (2011), 723–800.

    Article  MathSciNet  MATH  Google Scholar 

  35. T. Hytönen and A. McIntosh, Stability in p of the H∞-calculus of first-order systems in L p, Proc. Centre Math. Appl. Austral. Nat. Univ. 44 (2010), 167–181.

    Google Scholar 

  36. T. Hytönen, A. McIntosh, and P. Portal, Kato’s square root problem in Banach spaces, J. Funct. Anal. 254 (2008), 675–726.

    Article  MathSciNet  MATH  Google Scholar 

  37. T. Hytönen, A. McIntosh and P. Portal, Holomorphic functional calculus of Hodge-Dirac operators in L p, J. Evol. Equ. 11 (2011), 71–105.

    Article  MathSciNet  MATH  Google Scholar 

  38. T. Hytönen, J. van Neerven, and P. Portal, Conical square function estimates in UMD Banach spaces and applications to H∞-functional calculi, J. Anal. Math. 106 (2008), 317–351.

    Article  MathSciNet  MATH  Google Scholar 

  39. T. Hytönen, J. van Neerven, M. Veraar, and L. Weis, Analysis in Banach Spaces. Volume II: Probabilistic Methods and Operator Theory, Springer, 2017

    MATH  Google Scholar 

  40. N. Kalton and M. Mitrea, Stability results on interpolation scales of quasi-Banach spaces and applications, Trans. Amer. Math. Soc. 350 (1998), 3903–3922.

    Article  MathSciNet  MATH  Google Scholar 

  41. N. J. Kalton and L. Weis, The H∞-calculus and sums of closed operators, Math. Ann. 321 (2001), 319–345.

    Article  MathSciNet  MATH  Google Scholar 

  42. P. C. Kunstmann and L. Weis, Maximal L p Regularity for Parabolic Problems, Fourier Multiplier Theorems and H∞-functional Calculus, Springer-Verlag, 2004.

    MATH  Google Scholar 

  43. A. McIntosh, Operators which have an H∞ functional calculus, Proc. Centre Math. Appl. Austral. Nat. Univ. 14 (1986), 210–231.

    MathSciNet  MATH  Google Scholar 

  44. A. McIntosh and A. Nahmod, Heat kernel estimates and functional calculi of −bΔ, Math. Scand. 87 (2000), 287–319.

    Article  MathSciNet  MATH  Google Scholar 

  45. A. J. Morris. The Kato square root problem on submanifolds, J. Lond. Math. Soc. (2) 86 (2012), 879–910.

    Article  MathSciNet  MATH  Google Scholar 

  46. J. van Neerven, γ-radonifying operators–a survey, Proc. Centre Math. Appl. Austral. Nat. Univ. 44 (2010), 1–61.

    MathSciNet  MATH  Google Scholar 

  47. E. Stein. Interpolation of linear operators, Trans. Amer. Math. Soc. 83 (1956), 482–492.

    Article  MathSciNet  MATH  Google Scholar 

  48. E. M. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, 1970.

    MATH  Google Scholar 

Download references

Author information

Author notes

  1. Dorothee Frey

    Present address: Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

Authors and Affiliations

  1. Australian National University Mathematical Sciences Institute, John Dedman Building, Acton ACT, 2601, Australia

    Dorothee Frey & Pierre Portal

  2. Delft Institute of Applied Mathematics, Delft University of Technology, P.O. Box 5031, 2600, GA Delft, The Netherlands

    Alan McIntosh

Authors
  1. Dorothee Frey
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alan McIntosh
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Pierre Portal
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Dorothee Frey.

Additional information

Dedicated to the Memory of Alan Mcintosh (1942–2016)

The authors gratefully acknowledge support from the Australian Research Council through the Discovery Project DP120103692. This work is a key outcome of DP120103692.

Frey also acknowledges support from ARC DP110102488.

Portal is further supported by the ARC through the Future Fellowship FT130100607.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Frey, D., McIntosh, A. & Portal, P. Conical square function estimates and functional calculi for perturbed Hodge-Dirac operators in LP. JAMA 134, 399–453 (2018). https://doi.org/10.1007/s11854-018-0013-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11854-018-0013-3

Search

Navigation