Sobolev algebras on lie groups and riemannian manifolds

American Journal of Mathematics

Volumn 123, Issue 2, 2001, Pages 283-342

  Coulhon, Thierry ^a   Russ, Emmanuel ^a   Tardivel Nachef, Valérie ^b  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b AIX MARSEILLE UNIVERSITY   (France)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0035315549     PISSN: 00029327     EISSN: None     Source Type: Journal    
DOI: 10.1353/ajm.2001.0009     Document Type: Article

References (64)

1. D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Grundlehren Math. Wiss., vol. 314, Springer-Verlag, Berlin, 1996.
2. D. R. Adams and N. G. Meyers, Bessel potentials. Inclusion relations among classes of exceptional sets, Indiana Univ. Math. J. 22 (1973), 873-905.
3. G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack's inequalities and Riesz transforms on Lie groups with polynomial volume growth, Canad. J. Math. 44 (1992), 691-727.
4. N. Aronszajn and K. T. Smith, Theory of Bessel potentials I,Ann. Inst. Fourier 11 (1961), 385-475.
5. D. Bakry, Etude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités XXI, Lecture Notes in Math, vol. 1247, Springer-Verlag, 1987, pp. 137-172.
6. _, The Riesz transforms associated with second order differential operators, Seminar on Stochastic Processes 1988, Birkhäuser, Boston, 1989, pp. 1-43.
7. D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
8. O. Ben-Bassat, R. Strichartz, and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999), 197-217.
9. J. Bergh and J. Löfström, Interpolation spaces. An Introduction, Springer-Verlag, New York, 1976.
10. G. Bohnke, Algèbres de Sobolev sur certains groupes nilpotents, J. Funct. Anal. 63 (1985), 322-343.
11. N. Bouleau and F. Hirsch, Formes de Dirichlet générales et densité des variables aléatoires réelles sur l'espace de Wiener, J. Funct. Anal. 69 (1986), 229-259.
12. G. Bourdaud, Réalisation des espaces de Besov homogènes, Ark. Mat. 26 (1988), 41-54.
13. A. P. Calderón, Lebesgue spaces of differentiable functions and distributions, Proc. Sympos. Pure Math. 4 (1961), 33-49.
14. J. Cheeger, M. Gromov, and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator and the geometry of complete Riemannian manifolds, J. Differential Geom. 17 (1982), 15-53.
15. F. M. Christ and M. I. Weinstein, Dispersion of small-amplitude solutions of the generalized Korteweg-de Vries equation, J. Funct. Anal. 100 (1991), 87-109.
16. R. R. Coifman and Y. Meyer, Au-delà des opérateurs pseudo-différentiels, Astérisque 57 (1978).
17. _, Nonlinear harmonic analysis, operator theory and P. D. E., Beijing Lectures in Harmonic Analysis, Ann. of Math. Stud., vol. 112, Princeton University Press, Princeton, 1986, pp. 3-45.
18. T. Coulhon and L. Saloff-Coste, Semi-groupes d'opérateurs et espaces fonctionnels sur les groupes de Lie, J. Approx. Theory 65 (1991), 176-199.
19. J. Deny, Méthodes hilbertiennes en théorie du potentiel. Potential Theory, C.I.M.E., Cremonese, Rome, 1970.
20. P. Dorronsoro, A characterization of potential spaces, Proc. Amer. Math. Soc. 95 (1985), 21-31.
21. A. F. M. ter Elst, D. W. Robinson, and A. Sikora, Heat kernels and Riesz transforms on nilpotent Lie groups, Colloq. Math. 74 (1997), 191-218.
22. _, Riesz transforms and Lie groups of polynomial growth, J. Funct. Anal. 162 (1999), 14-51.
23. C. Fefferman, Inequalities for strongly singular convolution operators, Acta Math. 14 (1970), 9-36.
24. C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115.
25. G. B. Folland, Subelliptic estimates and function spaces on nilpotent Lie groups. Ark. Math. 13 (1975), 161-207.
26. G. B. Folland and E. M. Stein, Hardy Spaces on Homogeneous Groups, Princeton University Press and University of Tokyo Press, 1982.
27. M. Frazier and B. Jawerth, A discrete transform and decompositions of distribution spaces, J, Funct. Anal. 93 (1999), 34-170.
28. M. Fukushima, Y. Oshima, and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, De Gruyter, Berlin, 1994.
29. J. Garcia-Cuerva and J. L. Rubio de Francia, Weighted Norm Inequalities and Related Topics, North-Holland, Amsterdam, 1985.
30. A. Grigor'yan, On stochastically complete manifolds, Soviet Math. Dokl. 34 (1987).
31. Y Guivarc'h, Croissance polynomiale et périodes des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333-379.
32. A. Gulisashvili and M. A. Kon, Exact smoothing properties of Schrödinger semigroups, Amer. J. Math. 118 (1996), 1215-1248.
33. C. Herz, Lipschitz spaces and Bernstein's theorem on absolutely convergent Fourier transforms, J. Math. Mech. 18 (1968), 283-323.
34. T. Kato and G. Ponce, Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891-907.
35. H. Komatsu, Fractional powers of operators, Pacific J. Math. 19 (1966), 285-346.
36. _, Fractional powers of operators, VI: Interpolation of nonnegative operators and imbedding theorems, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 19 (1972), 1-62.
37. P. G. Lemarié, Algèbres d'opérateurs et semi-groupes de Poisson sur un espace de nature homogène, Publications Mathématiques d'Orsay, 1984, pp. 84-03.
38. P. Li and S. T. Yau, On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986), 153-201.
39. N. Lohoué and N. Th. Varopoulos, Remarques sur les transformées de Riesz sur les groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math. 301 (1985), 559-560.
40. V. G. Maz'ya and T. O. Shaposhnikova, Theory of Multipliers in Spaces of Differentiable Functions, Pitman, Boston, 1985.
41. S. Meda, On the Littlewood-Paley-Stein g-function, Trans. Amer. Math. Soc. 347 (1995), 2201-2212.
42. D. Morbidelli, Fractional Sobolev norms and structure of Carnot-Carathéodory balls for Hörmander vector fields, Studia Math. 139 (2000), 213-244.
43. A. Nagel, E. M. Stein, and S. Wainger, Balls and metrics defined by vector fields I: Basic properties, Acta Math. 155 (1985), 103-147.
44. J. Peetre, New Thoughts on Besov Spaces, Duke Univ. Math. Series, No. 1, 1976.
45. D. W. Robinson, Elliptic Operators and Lie Groups, Oxford Univerisity Press, 1991.
46. T. Runst and W. Sickel, Sobolev Spaces of Fractional Order, Nemytskij Operators and Nonlinear Partial Differential Equations, De Gruyter, Berlin, 1996.
47. K. Saka, Besov spaces and Sobolev spaces on a nilpotent Lie group, Tôhoku Math. J. 31 (1979), 383-437.
48. L. Saloff-Coste, Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat. 28 (1990), 315-331.
49. _, Parabolic Harnack inequality for divergence form second order differential operators, Potential Anal. 4 (1995), 429-467.
50. p,q < s < n/p, Ann. Mat. Pura Appl. 176 (1999), 209-250.
51. E. M. Stein, The characterization of functions arising as potentials, Bull. Amer. Math. Soc. 67 (1961), 102-104.
52. _, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, 1970.
53. _, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton University Press, 1970.
54. _, Harmonic Analysis: Real-variable Methods, Orthogonality, and Oscillatory Integrals, Princeton University Press, 1993.
55. E. M. Stein and G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Princeton University Press, 1975.
56. R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 1031-1060.
57. H. Triebel, Spaces of Besov-Hardy-Sobolev type on complete Riemannian manifolds, Ark. Mat. 24 (1986), 299-337.
58. _, Characterizations of function spaces on a complete Riemannian manifold with bounded geometry, Math. Nachr. 130 (1987), 321-346.
59. _, Function spaces on Lie groups, the Riemannian approach, J. London Math. Soc. 35 (1987), 327-338.
60. _, Theory of Function Spaces II, Birkhäuser, Boston, 1992.
61. N. Th. Varopoulos, Hardy-Littlewood theory for semi-groups, J. Funct. Anal. 63 (1985), 240-260.
62. _, Analysis on Lie groups, J. Funct. Anal. 76 (1988), 346-410.
63. _, Small time Gaussian estimates of heat diffusion kernels. Part I. The semi-group technique, Bull. Sci. Math. (2) 113 (1989), 253-277.
64. N. Th. Varopoulos, L. Saloff-Coste, and T. Coulhon, Analysis and Geometry on Groups, Cambridge University Press, 1992.