Skip to main content
SpringerLink
Log in

Riesz transform, Gaussian bounds and the method of wave equation

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL −α on L p for some α > 0. A very simple proof of the obtained result is based on the finite speed propagation property for the solution of the corresponding wave equation. We also discuss the relation between the Gaussian bounds and the finite speed propagation property. Using the wave equation methods we obtain a new natural form of the Gaussian bounds for the heat kernels for a large class of the generating operators. We describe a surprisingly elementary proof of the finite speed propagation property in a more general setting than it is usually considered in the literature.

As an application of the obtained results we prove boundedness of the Riesz transform on L p for all p ∈ (1,2] for Schrödinger operators with positive potentials and electromagnetic fields. In another application we discuss the Gaussian bounds for the Hodge Laplacian and boundedness of the Riesz transform on L p of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.

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.

Similar content being viewed by others

References

  1. Alexopoulos, G.: An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth. Canad. J. Math. 44(4), 691–727, 1992

    MATH  Google Scholar 

  2. Arendt, W., Grabosch, A., Greiner, G., Groh, U., Lotz, H. P., Moustakas, U., Nagel, R., Neubrander, F., Schlotterbeck, U.: One-parameter semigroups of positive operators. Springer-Verlag, Berlin, 1986

  3. Bakry, D.: The Riesz transforms associated with second order differential operators. In: Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), 17 of Progr. Probab. pages 1–43. Birkhäuser Boston, Boston, MA, 1989

  4. Cheeger, J., Gromov, M., Taylor, M.: Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds. J. Differential Geom. 17(1), 15–53 1982

    MATH  Google Scholar 

  5. Christ, M.: Lectures on singular integral operators. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990

  6. Coifman, R. R., Weiss, G.: Analyse harmonique non-commutative sur certains espaces homogènes. Springer-Verlag, Berlin, 1971. Étude de certaines intégrales singulières, Lecture Notes in Mathematics, Vol. 242

  7. Coulhon, T.: Heat kernels on non-compact Riemannian manifolds: a partial survey. In: Séminaire de Théorie Spectrale et Géométrie, No. 15, Année 1996–1997, 15 of Sémin. Théor. Spectr. Géom., pages 167–187. Univ. Grenoble I, Saint, 199?

  8. Coulhon, T.: Itération de Moser et estimation gaussienne du noyau de la chaleur. J. Operator Theory, 29(1), 157–165 1993

    Google Scholar 

  9. Coulhon, T., Duong, X. T.: Riesz transforms for 1≤ p ≤ 2. Trans. Amer. Math. Soc. 351(3), 1151–1169 1999

    Article  MATH  Google Scholar 

  10. Coulhon, T., Duong, X. T.: Riesz transforms for p > 2. C. R. Acad. Sci. Paris Sér. I Math. 332(11), 975–980 2001

    MATH  Google Scholar 

  11. Coulhon, T., Duong, X. T.: Riesz transform and related inequalities on non-compact Riemannian manifolds. Comm. Pure Appl. Math., 2002 to appear

  12. Davies, E. B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989

  13. Davies, E. B.: Heat kernel bounds, conservation of probability and the Feller property. J. Anal. Math. 58, 99–119, 1992. Festschrift on the occasion of the 70th birthday of Shmuel Agmon

    Google Scholar 

  14. Davies, E. B., Pang, M. M. H.: Sharp heat kernel bounds for some Laplace operators. Quart. J. Math. Oxford Ser. (2), 40(159), 281–290 1989

    Google Scholar 

  15. Dunford, N., Schwartz, J. T.: Linear operators. Part I. Wiley Classics Library. John Wiley & Sons Inc., New York, 1988. General theory, With the assistance of William G. Bade and Robert G. Bartle, Reprint of the 1958 original, A Wiley-Interscience Publication

  16. Duong, X. T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoamericana, 15(2), 233–265 1999

    Google Scholar 

  17. ter Elst, A. F. M., Robinson, D. W., Adam Sikora: Heat kernels and Riesz transforms on nilpotent Lie groups. Colloq. Math., 74(2), 191–218 1997

    Google Scholar 

  18. ter Elst, A. F. M., Robinson, D. W., Adam Sikora: Riesz transforms and Lie groups of polynomial growth. J. Funct. Anal., 162(1), 14–51 1999

    Google Scholar 

  19. Fefferman, Ch.: Inequalities for strongly singular convolution operators. Acta Math., 124, 9–36 1970

    Google Scholar 

  20. Folland, G. B.: Introduction to partial differential equations. Princeton University Press, Princeton, N.J., 1976. Preliminary informal notes of university courses and seminars in mathematics, Mathematical Notes

  21. Gaudry, G., Sjögren, P.: Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group. Math. Z., 232(2), 241–256 1999

    Google Scholar 

  22. Grigor′yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Differential Geom., 45(1), 33–52 1997

    Google Scholar 

  23. Grigor′yan, A.: Estimates of heat kernels on Riemannian manifolds. In Spectral theory and geometry (Edinburgh, 1998), 273 of London Math. Soc. Lecture Note Ser., pages 140–225. Cambridge Univ. Press, Cambridge, 1999

  24. Hebisch, W.: A multiplier theorem for Schrödinger operators. Colloq. Math., 60/61(2), 659–664 1990

    Google Scholar 

  25. Hörmander, L.: The analysis of linear partial differential operators. II. Springer- Verlag, Berlin, 1983. Differential operators with constant coefficients

  26. Hong-Quan Li: Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal., 161(1), 152–218 1999

    Google Scholar 

  27. Lohoué, N.: Transformées de Riesz et fonctions sommables. Amer. J. Math., 114(4), 875–922 1992

    Google Scholar 

  28. Markushevich, A. I.: Theory of functions of a complex variable. Vol. I, II, III. Chelsea Publishing Co., New York, English edition, 1977. Translated and edited by Richard A. Silverman

  29. Melrose, R.: Propagation for the wave group of a positive subelliptic second-order differential operator. In Hyperbolic equations and related topics (Katata/Kyoto, 1984), pages 181–192. Academic Press, Boston, MA, 1986

  30. Molchanov, S. A.: Diffusion processes, and Riemannian geometry. Uspehi Mat. Nauk, 30(1(181)), 3–59 1975

    Google Scholar 

  31. Ouhabaz, E. M.: L p boundedness of Riesz transform of magnetic Schrödinger operators. personal communication, 2000

  32. Robinson, D. W.: Elliptic operators and Lie groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991. Oxford Science Publications

  33. Saloff-Coste, L.: Analyse sur les groupes de Lie à croissance polynômiale. Ark. Mat., 28(2), 315–331 1990

    Google Scholar 

  34. Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble), 45(2), 513–546 1995

    Google Scholar 

  35. Shen, Z.: Estimates in L p for magnetic Schrödinger operators. Indiana Univ. Math. J., 45(3), 817–841 1996

    Google Scholar 

  36. Sikora, A.: Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2), 47(187), 371–382, 1996

    Google Scholar 

  37. Simon, B.: Maximal and minimal Schrödinger forms. J. Operator Theory, 1(1), 37–47 1979

    Google Scholar 

  38. Sjögren, P.: An estimate for a first-order Riesz operator on the affine group. Trans. Amer. Math. Soc., 351(8), 3301–3314 1999

    Google Scholar 

  39. Stein, E. M.: Some results in harmonic analysis in R n, for n→ ∞ . Bull. Amer. Math. Soc. (N.S.), 9(1), 71–73 1983

    Google Scholar 

  40. Stein, E. M.: Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. Princeton University Press, Princeton, NJ, 1993. With the assistance of Timothy S. Murphy, Monographs in Harmonic Analysis, III

  41. Stein, E. M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32

  42. Strichartz, R. S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal., 52(1), 48–79 1983

    Google Scholar 

  43. Taylor, M. E.: Partial differential equations. I, 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996. Basic theory

  44. Th, N., Varopoulos, L., Saloff-Coste, Coulhon, T.: Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992

  45. Warner, F. W.: Foundations of differentiable manifolds and Lie groups, 94 of Graduate Texts in Mathematics. Springer-Verlag, New York, 1983. Corrected reprint of the 1971 edition

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, New Mexico State University, Las Cruces, NM 88003-8001, USA

    Adam Sikora

Authors
  1. Adam Sikora
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Adam Sikora.

Additional information

Mathematics Subject Classification (1991): 42B20

The author was partially supported by Summer Research Award from New Mexico State University.

in final form: 8 June 2003

Rights and permissions

Reprints and permissions

About this article

Cite this article

Sikora, A. Riesz transform, Gaussian bounds and the method of wave equation. Math. Z. 247, 643–662 (2004). https://doi.org/10.1007/s00209-003-0639-3

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0639-3

Keywords

Search

Navigation