Mathematische Zeitschrift

, Volume 247, Issue 3, pp 643–662 | Cite as

Riesz transform, Gaussian bounds and the method of wave equation

Article

Abstract.

For an abstract self-adjoint operator L and a local operator A we study the boundedness of the Riesz transform AL−α on Lp 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 Lp 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 Lp of the Laplace-Beltrami operator on Riemannian manifolds for p > 2.

References

  1. 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, 1992MATHGoogle Scholar
  2. 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, 1986Google Scholar
  3. 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, 1989Google Scholar
  4. 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 1982MATHGoogle Scholar
  5. 5.
    Christ, M.: Lectures on singular integral operators. Published for the Conference Board of the Mathematical Sciences, Washington, DC, 1990Google Scholar
  6. 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 Google Scholar
  7. 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?Google Scholar
  8. 8.
    Coulhon, T.: Itération de Moser et estimation gaussienne du noyau de la chaleur. J. Operator Theory, 29(1), 157–165 1993Google Scholar
  9. 9.
    Coulhon, T., Duong, X. T.: Riesz transforms for 1≤ p ≤ 2. Trans. Amer. Math. Soc. 351(3), 1151–1169 1999CrossRefMATHGoogle Scholar
  10. 10.
    Coulhon, T., Duong, X. T.: Riesz transforms for p > 2. C. R. Acad. Sci. Paris Sér. I Math. 332(11), 975–980 2001MATHGoogle Scholar
  11. 11.
    Coulhon, T., Duong, X. T.: Riesz transform and related inequalities on non-compact Riemannian manifolds. Comm. Pure Appl. Math., 2002 to appearGoogle Scholar
  12. 12.
    Davies, E. B.: Heat kernels and spectral theory. Cambridge University Press, Cambridge, 1989Google Scholar
  13. 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 AgmonGoogle Scholar
  14. 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 1989Google Scholar
  15. 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 PublicationGoogle Scholar
  16. 16.
    Duong, X. T., McIntosh, A.: Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoamericana, 15(2), 233–265 1999Google Scholar
  17. 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 1997Google Scholar
  18. 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 1999Google Scholar
  19. 19.
    Fefferman, Ch.: Inequalities for strongly singular convolution operators. Acta Math., 124, 9–36 1970Google Scholar
  20. 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 NotesGoogle Scholar
  21. 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 1999Google Scholar
  22. 22.
    Grigor′yan, A.: Gaussian upper bounds for the heat kernel on arbitrary manifolds. J. Differential Geom., 45(1), 33–52 1997Google Scholar
  23. 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, 1999Google Scholar
  24. 24.
    Hebisch, W.: A multiplier theorem for Schrödinger operators. Colloq. Math., 60/61(2), 659–664 1990Google Scholar
  25. 25.
    Hörmander, L.: The analysis of linear partial differential operators. II. Springer- Verlag, Berlin, 1983. Differential operators with constant coefficientsGoogle Scholar
  26. 26.
    Hong-Quan Li: Estimations L p des opérateurs de Schrödinger sur les groupes nilpotents. J. Funct. Anal., 161(1), 152–218 1999Google Scholar
  27. 27.
    Lohoué, N.: Transformées de Riesz et fonctions sommables. Amer. J. Math., 114(4), 875–922 1992Google Scholar
  28. 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. SilvermanGoogle Scholar
  29. 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, 1986Google Scholar
  30. 30.
    Molchanov, S. A.: Diffusion processes, and Riemannian geometry. Uspehi Mat. Nauk, 30(1(181)), 3–59 1975Google Scholar
  31. 31.
    Ouhabaz, E. M.: L p boundedness of Riesz transform of magnetic Schrödinger operators. personal communication, 2000Google Scholar
  32. 32.
    Robinson, D. W.: Elliptic operators and Lie groups. Oxford Mathematical Monographs. The Clarendon Press Oxford University Press, New York, 1991. Oxford Science PublicationsGoogle Scholar
  33. 33.
    Saloff-Coste, L.: Analyse sur les groupes de Lie à croissance polynômiale. Ark. Mat., 28(2), 315–331 1990Google Scholar
  34. 34.
    Shen, Z.: L p estimates for Schrödinger operators with certain potentials. Ann. Inst. Fourier (Grenoble), 45(2), 513–546 1995Google Scholar
  35. 35.
    Shen, Z.: Estimates in L p for magnetic Schrödinger operators. Indiana Univ. Math. J., 45(3), 817–841 1996Google Scholar
  36. 36.
    Sikora, A.: Sharp pointwise estimates on heat kernels. Quart. J. Math. Oxford Ser. (2), 47(187), 371–382, 1996Google Scholar
  37. 37.
    Simon, B.: Maximal and minimal Schrödinger forms. J. Operator Theory, 1(1), 37–47 1979Google Scholar
  38. 38.
    Sjögren, P.: An estimate for a first-order Riesz operator on the affine group. Trans. Amer. Math. Soc., 351(8), 3301–3314 1999Google Scholar
  39. 39.
    Stein, E. M.: Some results in harmonic analysis in R n, for n→ ∞ . Bull. Amer. Math. Soc. (N.S.), 9(1), 71–73 1983Google Scholar
  40. 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, IIIGoogle Scholar
  41. 41.
    Stein, E. M., Weiss, G.: Introduction to Fourier analysis on Euclidean spaces. Princeton University Press, Princeton, N.J., 1971. Princeton Mathematical Series, No. 32Google Scholar
  42. 42.
    Strichartz, R. S.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal., 52(1), 48–79 1983Google Scholar
  43. 43.
    Taylor, M. E.: Partial differential equations. I, 115 of Applied Mathematical Sciences. Springer-Verlag, New York, 1996. Basic theoryGoogle Scholar
  44. 44.
    Th, N., Varopoulos, L., Saloff-Coste, Coulhon, T.: Analysis and geometry on groups. Cambridge University Press, Cambridge, 1992Google Scholar
  45. 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 editionGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Mexico State UniversityLas CrucesUSA

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