Journal of Fourier Analysis and Applications

, Volume 18, Issue 5, pp 995–1066 | Cite as

Heat Kernel Generated Frames in the Setting of Dirichlet Spaces

Article

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.

Keywords

Heat kernel Gaussian bounds Functional calculus Sampling Frames Besov spaces 

Mathematics Subject Classification

58J35 42C15 43A85 46E35 

References

  1. 1.
    Anger, B., Lembcke, J.: Hahn-Banach type theorems for hypolinear functionals on preordered topological vector spaces. Pac. J. Math. 54, 13–33 (1974) MathSciNetMATHGoogle Scholar
  2. 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) CrossRefGoogle Scholar
  3. 3.
    Bergh, L., Löfström, J.: Interpolation Spaces. Springer, Berlin (1976) MATHCrossRefGoogle Scholar
  4. 4.
    Butzer, P., Berens, H.: Semi-Groups of Operators and Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 145. Springer, New York (1967) MATHCrossRefGoogle Scholar
  5. 5.
    Biroli, M., Mosco, U.: A Saint-Venant type principle for Dirichlet forms on discontinuous media. Ann. Mat. Pura Appl. CLXIX, 125–181 (1995) MathSciNetCrossRefGoogle Scholar
  6. 6.
    Biroli, M., Mosco, U.: Sobolev inequalities on homogeneous spaces. Potential Anal. 4, 311–325 (1995) MathSciNetMATHCrossRefGoogle Scholar
  7. 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) MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    Bouleau, N., Hirsch, F.: Dirichlet Forms and Analysis on Wiener Space. de Gruyter, Berlin (1991) MATHCrossRefGoogle Scholar
  9. 9.
    Carron, G., Ouhabaz, E.-M., Coulhon, T.: Gaussian estimates and L p-boundedness of Riesz means. J. Evol. Equ. 2, 299–317 (2002) MathSciNetCrossRefGoogle Scholar
  10. 10.
    Coifman, R., Weiss, G.: Analyse Harmonique Non-commutative sur Certains Espaces Homogènes. Lecture Notes in Math., vol. 242. Springer, Berlin (1971) MATHGoogle Scholar
  11. 11.
    Coifman, R., Weiss, G.: Extensions of Hardy spaces and their use in analysis. Bull. Am. Math. Soc. 83, 569–645 (1977) MathSciNetMATHCrossRefGoogle Scholar
  12. 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) MathSciNetMATHCrossRefGoogle Scholar
  13. 13.
    Coulhon, T., Sikora, A.: Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem. Proc. Lond. Math. Soc. 96, 507–544 (2008) MathSciNetMATHCrossRefGoogle Scholar
  14. 14.
    Davies, E.B.: Heat Kernels and Spectral Theory. Cambridge University Press, Cambridge (1989) MATHCrossRefGoogle Scholar
  15. 15.
    DeVore, R., Lorentz, G.G.: Constructive Approximation. Grundlehren der Mathematischen Wissenschaften, vol. 303. Springer, Berlin (1993) MATHGoogle Scholar
  16. 16.
    Dunford, N., Schwartz, J.: Linear Operators. Part I: General Theory. Wiley-Interscience, New York (1958) Google Scholar
  17. 17.
    Duong, X.T., Ouhabaz, E.-M., Sikora, A.: Plancherel-type estimates and sharp spectral multipliers. J. Funct. Anal. 196(2), 443–485 (2002) MathSciNetCrossRefGoogle Scholar
  18. 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) MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Folland, G.: Real Analysis. Modern Techniques and Their Applications, 2nd edn. Wiley-Interscience, New York (1999) MATHGoogle Scholar
  20. 20.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. De Gruyter Studies in Mathematics, vol. 19. De Gruyter, Berlin (1994) MATHCrossRefGoogle Scholar
  21. 21.
    Frazier, M., Jawerth, B.: Decomposition of Besov spaces. Indiana Univ. Math. J. 34, 777–799 (1985) MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Frazier, M., Jawerth, B.: A discrete transform and decomposition of distribution spaces. J. Funct. Anal. 93, 34–170 (1990) MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    Frazier, M., Jawerth, B., Weiss, G.: Littlewood-Paley theory and the study of function spaces. CBMS No. 79. AMS (1991) Google Scholar
  24. 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 MathSciNetMATHCrossRefGoogle Scholar
  25. 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) MATHGoogle Scholar
  26. 26.
    Grigor’yan, A.: Heat Kernel and Analysis on Manifolds. AMS/IP Studies in Advanced Mathematics, vol. 47 (2009) MATHGoogle Scholar
  27. 27.
    Gyrya, P., Saloff-Coste, L.: Neumann and Dirichlet Heat Kernels in Inner Uniform Domains. Astérisque, vol. 336. Société Mathématique de France, Paris (2011) MATHGoogle Scholar
  28. 28.
    Hebisch, W., Saloff-Coste, L.: On the relation between elliptic and parabolic Harnack inequalities. Ann. Inst. Fourier 51(5), 1437–1481 (2001) MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Ivanov, K., Petrushev, P., Xu, Y.: Sub-exponentially localized kernels and frames induced by orthogonal expansions. Math. Z. 264, 361–397 (2010) MathSciNetMATHCrossRefGoogle Scholar
  30. 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. 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) MathSciNetMATHCrossRefGoogle Scholar
  32. 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) MathSciNetMATHCrossRefGoogle Scholar
  33. 33.
    Kerkyacharian, G., Petrushev, P.: Heat kernel based decomposition of spaces of distributions in the framework of Dirichlet spaces. Preprint Google Scholar
  34. 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) MathSciNetMATHCrossRefGoogle Scholar
  35. 35.
    Kyriazis, G., Petrushev, P., Xu, Y.: Jacobi decomposition of weighted Triebel-Lizorkin and Besov spaces. Stud. Math. 186, 161–202 (2008) MathSciNetMATHCrossRefGoogle Scholar
  36. 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) MathSciNetMATHCrossRefGoogle Scholar
  37. 37.
    Lemarié, P.: Base d’ondelettes sur les groupes de Lie stratifiés. Bull. Soc. Math. Fr. 117, 211–232 (1989) MATHGoogle Scholar
  38. 38.
    Li, P., Yau, S.T.: On the parabolic kernel of the Schrödinger operator. Acta Math. 156, 153–201 (1986) MathSciNetCrossRefGoogle Scholar
  39. 39.
    Maheux, P.: Estimations du noyau de la chaleur sur les espaces homogènes. J. Geom. Anal. 8, 65–96 (1998) MathSciNetMATHCrossRefGoogle Scholar
  40. 40.
    Moser, J.: A Harnack inequality for parabolic differential equations. Commun. Pure Appl. Math. 17, 101–134 (1964). Corrigendum 20, 232–236 (1967) MATHCrossRefGoogle Scholar
  41. 41.
    Moser, J.: On a pointwise estimate for parabolic differential equations. Commun. Pure Appl. Math. 24, 727–740 (1971) MATHCrossRefGoogle Scholar
  42. 42.
    Narcowich, F., Petrushev, P., Ward, J.: Localized tight frames on spheres. SIAM J. Math. Anal. 38, 574–594 (2006) MathSciNetCrossRefGoogle Scholar
  43. 43.
    Narcowich, F., Petrushev, P., Ward, J.: Decomposition of Besov and Triebel-Lizorkin spaces on the sphere. J. Funct. Anal. 238, 530–564 (2006) MathSciNetMATHCrossRefGoogle Scholar
  44. 44.
    Nash, J.: Continuity of solutions of parabolic and elliptic equations. Am. J. Math. 80, 931–954 (1958) MathSciNetMATHCrossRefGoogle Scholar
  45. 45.
    Ouhabaz, E.M.: Analysis of Heat Equations on Domains. London Mathematical Society Monographs Series, vol. 31. Princeton University Press, Princeton (2005) MATHGoogle Scholar
  46. 46.
    Peetre, J.: New Thoughs on Besov Spaces. Duke University (1976) Google Scholar
  47. 47.
    Petrushev, P., Xu, Y.: Localized polynomial frames on the interval with Jacobi weights. J. Fourier Anal. Appl. 11, 557–575 (2005) MathSciNetMATHCrossRefGoogle Scholar
  48. 48.
    Petrushev, P., Xu, Y.: Localized polynomial frames on the ball. Constr. Approx. 27, 121–148 (2008) MathSciNetMATHCrossRefGoogle Scholar
  49. 49.
    Petrushev, P., Xu, Y.: Decomposition of spaces of distributions induced by Hermite expansions. J. Fourier Anal. Appl. 14, 372–414 (2008) MathSciNetMATHCrossRefGoogle Scholar
  50. 50.
    Robinson, D.W.: Elliptic Operators on Lie Groups. Oxford University Press, London (1991) Google Scholar
  51. 51.
    Saloff-Coste, L.: A note on Poincaré, Sobolev and Harnack inequalities. Duke Math. J. 65(IMRN), 27–38 (1992) MathSciNetGoogle Scholar
  52. 52.
    Saloff-Coste, L.: Parabolic Harnack inequality for divergence form second order differential operators. Potential Anal. 4(4), 429–467 (1995) MathSciNetMATHCrossRefGoogle Scholar
  53. 53.
    Saloff-Coste, L.: Aspects of Sobolev-Type Inequalities. London Mathematical Society Lecture Note Series, vol. 289. Cambridge University Press, Cambridge (2002) MATHGoogle Scholar
  54. 54.
    Saloff-Coste, L., Stroock, D.W.: Opérateurs uniformément sous-elliptiques sur les groupes de Lie. J. Funct. Anal. 98(1), 97–121 (1991) MathSciNetMATHCrossRefGoogle Scholar
  55. 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) MathSciNetMATHGoogle Scholar
  56. 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) MathSciNetMATHGoogle Scholar
  57. 57.
    Sturm, K.T.: Analysis on local Dirichlet spaces. III. The parabolic Harnack inequality. J. Math. Pures Appl. 75, 273–297 (1998) MathSciNetGoogle Scholar
  58. 58.
    Szegö, G.: Orthogonal Polynomials, 4th edn. Amer. Math. Soc. Colloq. Publ., vol. 23. Am. Math. Soc., Providence (1975) MATHGoogle Scholar
  59. 59.
    Triebel, H.: In: Interpolation Theory, Function Spaces, Differential Operators. North-Holland Mathematical Library, vol. 18. North-Holland, Amsterdam (1978) Google Scholar
  60. 60.
    Triebel, H.: Theory of Function Spaces. Monographs in Math., vol. 78. Birkhäuser, Basel (1983) CrossRefGoogle Scholar
  61. 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. 62.
    Yosida, K.: Functional Analysis. Springer, Berlin (1980) MATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  1. 1.Equipe AGM, CNRS-UMR 8088Université de Cergy-PontoiseCergy-PontoiseFrance
  2. 2.Laboratoire de Probabilités et Modèles Aléatoires, CNRS-UMR 7599Université Paris VI et Université Paris VIIParisFrance
  3. 3.Department of MathematicsUniversity of South CarolinaColumbiaUSA

Personalised recommendations