Skip to main content
SpringerLink
Log in

Heat Kernel Generated Frames in the Setting of Dirichlet Spaces

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

Abstract

Wavelet bases and frames consisting of band limited functions of nearly exponential localization on ℝd are a powerful tool in harmonic analysis by making various spaces of functions and distributions more accessible for study and utilization, and providing sparse representation of natural function spaces (e.g. Besov spaces) on ℝd. Such frames are also available on the sphere and in more general homogeneous spaces, on the interval and ball. The purpose of this article is to develop band limited well-localized frames in the general setting of Dirichlet spaces with doubling measure and a local scale-invariant Poincaré inequality which lead to heat kernels with small time Gaussian bounds and Hölder continuity. As an application of this construction, band limited frames are developed in the context of Lie groups or homogeneous spaces with polynomial volume growth, complete Riemannian manifolds with Ricci curvature bounded from below and satisfying the volume doubling property, and other settings. The new frames are used for decomposition of Besov spaces in this general setting.

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. Anger, B., Lembcke, J.: Hahn-Banach type theorems for hypolinear functionals on preordered topological vector spaces. Pac. J. Math. 54, 13–33 (1974)

    MathSciNet  MATH  Google Scholar 

  2. Albeverio, S.: Theory of Dirichlet forms and applications. In: Lectures on Probability Theory and Statistics, Saint-Flour, 2000. Lecture Notes in Math., vol. 1816, pp. 1–106. Springer, Berlin (2003) (English summary)

    Chapter  Google Scholar 

  3. Bergh, L., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976)

    Book  MATH  Google Scholar 

  4. Butzer, P., Berens, H.: Semi-Groups of Operators and Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 145. Springer, New York (1967)

    Book  MATH  Google Scholar 

  5. Biroli, M., Mosco, U.: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. CLXIX, 125–181 (1995)

    Article  MathSciNet  Google Scholar 

  6. Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Potential Anal. 4, 311–325 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bui, H.-Q., Duong, X.T., Yan, L.: Calderón reproducing formulas and new besov spaces associated with operators. Adv. Math. 229, 2449–2502 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  8. Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. de Gruyter, Berlin (1991)

    Book  MATH  Google Scholar 

  9. Carron, G., Ouhabaz, E.-M., Coulhon, T.: Gaussian estimates and L p-boundedness of Riesz means. J. Evol. Equ. 2, 299–317 (2002)

    Article  MathSciNet  Google Scholar 

  10. Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math., vol. 242. Springer, Berlin (1971)

    MATH  Google Scholar 

  11. Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  12. Coulhon, T., Saloff-Coste, L.: Semi-groupes d’opérateurs et espaces fonctionnels sur les groupes de Lie. J. Approx. Theory 65(2), 176–199 (1991)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989)

    Book  MATH  Google Scholar 

  15. DeVore, R., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 303. Springer, Berlin (1993)

    MATH  Google Scholar 

  16. Dunford, N., Schwartz, J.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York (1958)

    Google Scholar 

  17. Duong, X.T., Ouhabaz, E.-M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2), 443–485 (2002)

    Article  MathSciNet  Google Scholar 

  18. Ferreira, J.C., Menegatto, V.A.: Eigenvalues of integral operators defined by smooth positive definite kernels. Integral Equ. Oper. Theory 64, 61–81 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  19. Folland, G.: Real Analysis. Modern Techniques and Their Applications, 2nd edn. Wiley-Interscience, New York (1999)

    MATH  Google Scholar 

  20. Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics, vol. 19. De Gruyter, Berlin (1994)

    Book  MATH  Google Scholar 

  21. Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985)

    Article  MathSciNet  MATH  Google Scholar 

  22. Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  23. Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces. CBMS No. 79. AMS (1991)

  24. Geller, D., Pesenson, I.Z.: Band-limited localized Parseval frames and Besov spaces on compact homogeneous manifolds. J. Geom. Anal. 21, 334–371 (2011). doi:10.1007/s12220-010-9150-3

    Article  MathSciNet  MATH  Google Scholar 

  25. Grigor’yan, A.: Heat kernel on non-compact manifolds. Mat. Sb. 182, 55–87 (1991) (Russian). Mat. USSR Sb. 72, 47–77 (1992) (English)

    MATH  Google Scholar 

  26. Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. AMS/IP Studies in Advanced Mathematics, vol. 47 (2009)

    MATH  Google Scholar 

  27. Gyrya, P., Saloff-Coste, L.: Neumann and Dirichlet Heat Kernels in Inner Uniform Domains. Astérisque, vol. 336. Société Mathématique de France, Paris (2011)

    MATH  Google Scholar 

  28. Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier 51(5), 1437–1481 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  29. Ivanov, K., Petrushev, P., Xu, Y.: Sub-exponentially localized kernels and frames induced by orthogonal expansions. Math. Z. 264, 361–397 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  30. Ivanov, K., Petrushev, P., Xu, Y.: Decomposition of spaces of distributions induced by tensor product bases. J. Funct. Anal. (to appear). doi:10.1016/j.jfa.2012.06.006

  31. Kerkyacharian, G., Kyriazis, G., Le Pennec, E., Petrushev, P., Picard, D.: Inversion of noisy radon transform by SVD based needlet. Appl. Comput. Harmon. Anal. 28, 24–45 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  32. Kerkyacharian, G., Kyriazis, G., Petrushev, P., Picard, D., Willer, T.: Needlet algorithms for estimation in inverse problems. Electron. J. Stat. 1, 30–76 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  33. Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Preprint

  34. Kerkyacharian, G., Petrushev, P., Picard, D., Xu, Y.: Decomposition of Triebel-Lizorkin and Besov spaces in the context of Laguerre expansions. J. Funct. Anal. 256, 1137–1188 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  35. Kyriazis, G., Petrushev, P., Xu, Y.: Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces. Stud. Math. 186, 161–202 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  36. Kyriazis, G., Petrushev, P., Xu, Y.: Decomposition of weighted Triebel-Lizorkin and Besov spaces on the ball. Proc. Lond. Math. Soc. 97, 477–513 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  37. Lemarié, P.: Base d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. Fr. 117, 211–232 (1989)

    MATH  Google Scholar 

  38. Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986)

    Article  MathSciNet  Google Scholar 

  39. Maheux, P.: Estimations du noyau de la chaleur sur les espaces homogènes. J. Geom. Anal. 8, 65–96 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  40. Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964). Corrigendum 20, 232–236 (1967)

    Article  MATH  Google Scholar 

  41. Moser, J.: On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971)

    Article  MATH  Google Scholar 

  42. Narcowich, F., Petrushev, P., Ward, J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006)

    Article  MathSciNet  Google Scholar 

  43. Narcowich, F., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958)

    Article  MathSciNet  MATH  Google Scholar 

  45. Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005)

    MATH  Google Scholar 

  46. Peetre, J.: New Thoughs on Besov Spaces. Duke University (1976)

    Google Scholar 

  47. Petrushev, P., Xu, Y.: Localized polynomial frames on the interval with Jacobi weights. J. Fourier Anal. Appl. 11, 557–575 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  48. Petrushev, P., Xu, Y.: Localized polynomial frames on the ball. Constr. Approx. 27, 121–148 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  49. Petrushev, P., Xu, Y.: Decomposition of spaces of distributions induced by Hermite expansions. J. Fourier Anal. Appl. 14, 372–414 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  50. Robinson, D.W.: Elliptic Operators on Lie Groups. Oxford University Press, London (1991)

    Google Scholar 

  51. Saloff-Coste, L.: A note on Poincaré, Sobolev and Harnack inequalities. Duke Math. J. 65(IMRN), 27–38 (1992)

    MathSciNet  Google Scholar 

  52. Saloff-Coste, L.: Parabolic Harnack inequality for divergence form second order differential operators. Potential Anal. 4(4), 429–467 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  53. Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002)

    MATH  Google Scholar 

  54. Saloff-Coste, L., Stroock, D.W.: Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98(1), 97–121 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  55. Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and L p -Liouville properties. J. Reine Angew. Math. 456, 173–196 (1994)

    MathSciNet  MATH  Google Scholar 

  56. Sturm, K.T.: Analysis on local Dirichlet spaces. II. Upper Gaussian estimates for the fundamental solutions of parabolic equations. Osaka J. Math. 32, 275–312 (1995)

    MathSciNet  MATH  Google Scholar 

  57. Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1998)

    MathSciNet  Google Scholar 

  58. Szegö, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1975)

    MATH  Google Scholar 

  59. Triebel, H.: In: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam (1978)

    Google Scholar 

  60. Triebel, H.: Theory of Function Spaces. Monographs in Math., vol. 78. Birkhäuser, Basel (1983)

    Book  Google Scholar 

  61. Varopoulos, N., Saloff-Coste, L., Coulhon, T.: Analysis and Geometry on Groups. Cambridge Tracts in Mathematics, vol. 100. Cambridge University Press, Cambridge (1992)

    Google Scholar 

  62. Yosida, K.: Functional Analysis. Springer, Berlin (1980)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Equipe AGM, CNRS-UMR 8088, Université de Cergy-Pontoise, Cergy-Pontoise, France

    T. Coulhon

  2. Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599, Université Paris VI et Université Paris VII, rue de Clisson, 75013, Paris, France

    G. Kerkyacharian

  3. Department of Mathematics, University of South Carolina, Columbia, SC, 29208, USA

    P. Petrushev

Authors
  1. T. Coulhon
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. G. Kerkyacharian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. P. Petrushev
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to P. Petrushev.

Additional information

Communicated by Hans G. Feichtinger.

P. Petrushev has been supported by NSF Grant DMS-1211528.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Coulhon, T., Kerkyacharian, G. & Petrushev, P. Heat Kernel Generated Frames in the Setting of Dirichlet Spaces. J Fourier Anal Appl 18, 995–1066 (2012). https://doi.org/10.1007/s00041-012-9232-7

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00041-012-9232-7

Keywords

Mathematics Subject Classification

Search

Navigation