Littlewood-Paley-Stein functions on complete Riemannian manifolds for 1 ≤ p ≤ 2

Studia Mathematica

Volumn 154, Issue 1, 2003, Pages 37-57

  Coulhon, Thierry ^a   Duong, Xuan Think ^b   Li, Xiang Dong ^c  

^a UNIVERSITÉ DE CERGY PONTOISE   (France)

^b MACQUARIE UNIVERSITY   (Australia)

^c UNIVERSITY OF OXFORD   (United Kingdom)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0037224674     PISSN: 00393223     EISSN: None     Source Type: Journal    
DOI: 10.4064/sm154-1-4     Document Type: Article

References (41)

1. W. Arendt and A. F. M. ter Elst, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory 38 (1997), 87-130.
2. P. Auscher, Regularity theorems and heat kernels for elliptic operators, J. London Math. Soc. 54 (1996), 284-296.
3. P. Auscher, X. T. Duong and A. McIntosh, Boundedness of Banach space valued singular integral operators and Hardy spaces, preprint.
4. n, Ann. of Math., to appear.
5. P. Auscher, A. McIntosh and Ph. Tchamitchian, Heat kernels of second order complex elliptic operators and applications, J. Funct. Anal. 152 (1998), 22-73.
6. 2 theory, J. Anal. Math., to appear.
7. D. Bakry, Transformations de Riesz pour les semi-groupes symétriques, in: Séminaire de Probabilités XIX, Lecture Notes in Math. 1123, Springer, 1985, 130-175.
8. _, Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, in: Séminaire de Probabilités XXI, Lecture Notes in Math. 1247, Springer, 1987, 137-172.
9. _, The Riesz transforms associated with second order differential operators, in: Seminar on Stochastic Processes, 1988 (Gainesville, FL, 1988), Progr. Probab. 17, Birkhaüser, 1989, 1-43.
10. S. Blunck and P. Kunstmann, Weak type (p,p) estimates for Riesz transforms, preprint.
11. J. C. Chen, Heat kernels on positively curved manifolds and applications, Ph.D. thesis, Hangzhou Univ., 1987.
12. L. Chevalier, private communication.
13. A. Chojnowska-Michalik and B. Gołdys, Generalized Ornstein-Uhlenbeck semi-groups: Littlewood-Paley-Stein inequalities and the P. A. Meyer equivalence of norms, J. Funct. Anal. 182 (2001), 243-279.
14. T. Coulhon and X. T. Duong, Riesz transforms for 1 ≤ p ≤ 2, Trans. Amer. Math. Soc. 351 (1999), 1151-1169.
15. _, _, Maximal regularity and kernel bounds: observations on a theorem by Hieber and Prüss, Adv. Differential Equations 5 (2000), 343-368.
16. _, _, Riesz transforms for p > 2, C. R. Acad. Sci. Paris Sér. I 332 (2001), 975-980.
17. _, _, Riesz transform and related inequalities on non-compact Riemannian manifolds, preprint.
18. E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Univ. Press, 1989.
19. _, Non-Gaussian aspects of heat kernel behaviour, J. London Math. Soc. (2) 55 (1997), 105-125.
20. X. T. Duong and A. McIntosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), 233-265.
21. p boundedness of Riesz transforms associated with divergence form operators, in: Workshop on Analysis and Applications (Brisbane, 1997), Proc. Centre Math. Appl. Austral. Nat. Univ. 37, Austral. Nat. Univ., Canberra, 1999, 15-25.
22. X. T. Duong and D. Robinson, Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal. 142 (1996), 89-128.
23. A. Grigor'yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), 395-452.
24. _, Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995), 363-389.
25. _, Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Differential Geom. 45 (1997), 33-52.
26. Y. Han and E. Sawyer, Littlewood-Paley theory on spaces of homogeneous type and the classical function spaces, Mem. Amer. Math. Soc. 110 (1994).
27. T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Springer, 1976.
28. X. D. Li, Riesz transforms and Schrödinger operators on complete Riemannian manifolds with negative Ricci curvature, preprint.
29. N. Lohoué, Estimation des fonctions de Littlewood-Paley-Stein sur les variétés riemanniennes à courbure non positive, Ann. Sci. École Norm. Sup. 20 (1987), 505-544.
30. _, Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables, C. R. Acad. Sci. Paris 306 (1988), 327-330.
31. S. Meda, On the Littlewood-Paley-Stein g-function, Trans. Amer. Math. Soc. 347 (1995), 2201-2212.
32. P.-A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, exposé II; les inégalités générales, in: Séminaire de Probabilités X, Lecture Notes in Math. 511, Springer, 1976, 165-174.
33. _, Retour sur la théorie de Littlewood-Paley, in: Séminaire de Probabilités XV, Lecture Notes in Math. 850, Springer, 1981, 151-166.
34. 1-BMO duality on Riemannian manifolds, Dissertationes Math., to appear.
35. I. Shigekawa, Littlewood-Paley inequality for a diffusion satisfying the logarithmic Sobolev inequality and for the Brownian motion on a Riemannian manifold with boundary, preprint.
36. I. Shigekawa and N. Yoshida, Littlewood-Paley-Stein inequality for a symmetric diffusion, J. Math. Soc. Japan 44 (1992), 251-280.
37. E. Stein, On the functions of Littlewood-Paley, Lusin and Marcinkiewicz, Trans. Amer. Math. Soc. 88 (1958), 137-174.
38. _, Topics in Harmonic Analysis Related to the Littlewood-Paley Theory, Princeton Univ. Press, 1970.
39. _, Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, 1970.
40. N. Yoshida, The Littlewood-Paley-Stein inequality on an infinite-dimensional manifold, J. Funct. Anal. 122 (1994), 402-427.
41. _, Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ. 32 (1992), 621-654.