Riesz transform on manifolds and heat kernel regularity

Pascal Auscher; Thierry Coulhon; Xuan Thinh Duong; Steve Hofmann

Annales scientifiques de l'École Normale Supérieure (2004)

  • Volume: 37, Issue: 6, page 911-957
  • ISSN: 0012-9593

How to cite


Auscher, Pascal, et al. "Riesz transform on manifolds and heat kernel regularity." Annales scientifiques de l'École Normale Supérieure 37.6 (2004): 911-957. <http://eudml.org/doc/82649>.

author = {Auscher, Pascal, Coulhon, Thierry, Duong, Xuan Thinh, Hofmann, Steve},
journal = {Annales scientifiques de l'École Normale Supérieure},
keywords = {Riesz transform; Sobolev space; Hodge decomposition},
language = {eng},
number = {6},
pages = {911-957},
publisher = {Elsevier},
title = {Riesz transform on manifolds and heat kernel regularity},
url = {http://eudml.org/doc/82649},
volume = {37},
year = {2004},

AU - Auscher, Pascal
AU - Coulhon, Thierry
AU - Duong, Xuan Thinh
AU - Hofmann, Steve
TI - Riesz transform on manifolds and heat kernel regularity
JO - Annales scientifiques de l'École Normale Supérieure
PY - 2004
PB - Elsevier
VL - 37
IS - 6
SP - 911
EP - 957
LA - eng
KW - Riesz transform; Sobolev space; Hodge decomposition
UR - http://eudml.org/doc/82649
ER -


  1. [1] Alexopoulos G., An application of homogeneization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math.44 (4) (1992) 691-727. Zbl0792.22005MR1178564
  2. [2] AuscherP., On necessary and sufficient conditions for L p -estimates of Riesz transforms associated to elliptic operators on R n and related estimates, preprint 2004-04, Université de Paris-Sud, Mathématiques. 
  3. [3] Auscher P., Hofmann S., Lacey M., McIntosh A., Tchamitchian P., The solution of the Kato square root problem for second order elliptic operators on R n , Annals of Math.156 (2002) 633-654. Zbl1128.35316MR1933726
  4. [4] Auscher P., Tchamitchian P., Square Root Problem for Divergence Operators and Related Topics, Astérisque, vol. 249, 1998. Zbl0909.35001MR1651262
  5. [5] Auscher P., Coulhon T., Riesz transforms on manifolds and Poincaré inequalities, preprint, 2004. Zbl1116.58023MR2185868
  6. [6] Bakry D., Transformations de Riesz pour les semi-groupes symétriques, Seconde partie: étude sous la condition Γ 2 0 , in: Séminaire de Probabilités XIX, Lecture Notes, vol. 1123, Springer, Berlin, 1985, pp. 145-174. Zbl0561.42011MR889473
  7. [7] Bakry D., Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée, in: Séminaire de Probabilités XXI, Lecture Notes, vol. 1247, Springer, Berlin, 1987, pp. 137-172. Zbl0629.58018MR941980
  8. [8] Bakry D., The Riesz transforms associated with second order differential operators, in: Seminar on Stochastic Processes, vol. 88, Birkhäuser, Basel, 1989. Zbl0689.58032MR990472
  9. [9] Blunck S., Kunstmann P., Calderón–Zygmund theory for non-integral operators and the H functional calculus, Rev. Mat. Iberoamer.19 (3) (2003) 919-942. Zbl1057.42010MR2053568
  10. [10] Caffarelli L., Peral I., On W 1 , p estimates for elliptic equations in divergence form, Comm. Pure Appl. Math.51 (1998) 1-21. Zbl0906.35030MR1486629
  11. [11] Carron G., Formes harmoniques L 2 sur les variétés non-compactes, Rend. Mat. Appl.7 (21) (2001), 1–4, 87–119. Zbl1049.58006MR1884938
  12. [12] Chavel I., Riemannian Geometry: A Modern Introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. Zbl0810.53001MR1271141
  13. [13] Chen J.-C., Heat kernels on positively curved manifolds and applications, Ph. D. thesis, Hanghzhou university, 1987. 
  14. [14] Coifman R., Rochberg R., Another characterization of BMO, Proc. Amer. Math. Soc.79 (2) (1980) 249-254. Zbl0432.42016MR565349
  15. [15] Coifman R., Weiss G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc.83 (1977) 569-645. Zbl0358.30023MR447954
  16. [16] Coulhon T., Noyau de la chaleur et discrétisation d'une variété riemannienne, Israel J. Math.80 (1992) 289-300. Zbl0772.58055MR1202573
  17. [17] Coulhon T., Off-diagonal heat kernel lower bounds without Poincaré, J. London Math. Soc.68 (3) (2003) 795-816. Zbl1083.58025MR2010012
  18. [18] Coulhon T., Duong X.T., Riesz transforms for 1 p 2 , Trans. Amer. Math. Soc.351 (1999) 1151-1169. Zbl0973.58018MR1458299
  19. [19] Coulhon T., Duong X.T., Riesz transforms for p g t ; 2 , CRAS Paris, série I332 (11) (2001) 975-980. Zbl0987.43001MR1838122
  20. [20] Coulhon T., Duong X.T., Riesz transform and related inequalities on non-compact Riemannian manifolds, Comm. Pure Appl. Math.56 (12) (2003) 1728-1751. Zbl1037.58017MR2001444
  21. [21] Coulhon T., Grigor'yan A., Pittet Ch., A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier51 (6) (2001) 1763-1827. Zbl1137.58307MR1871289
  22. [22] Coulhon T., Li H.Q., Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz, Archiv der Mathematik83 (2004) 229-242. Zbl1076.58017MR2108551
  23. [23] Coulhon T., Müller D., Zienkiewicz J., About Riesz transforms on the Heisenberg groups, Math. Ann.305 (2) (1996) 369-379. Zbl0859.22006MR1391221
  24. [24] Coulhon T., Saloff-Coste L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamer.9 (2) (1993) 293-314. Zbl0782.53066MR1232845
  25. [25] Coulhon T., Sikora A., Gaussian heat kernel bounds via Phragmén–Lindelöf theorems, preprint. Zbl1148.35009
  26. [26] Cranston M., Gradient estimates on manifolds using coupling, J. Funct. Anal.99 (1) (1991) 110-124. Zbl0770.58038MR1120916
  27. [27] Davies E.B., Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math.58 (1992) 99-119. Zbl0808.58041MR1226938
  28. [28] Davies E.B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal.132 (1995) 141-169. Zbl0839.35034MR1346221
  29. [29] De Rham G., Variétés différentiables, formes, courants, formes harmoniques, Hermann, Paris, 1973. Zbl0065.32401MR346830
  30. [30] Dragičević O., Volberg A., Bellman functions and dimensionless estimates of Riesz transforms, preprint. Zbl1114.42006
  31. [31] Driver B., Melcher T., Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal., appeared online 11 September 2004. Zbl1071.22005MR2124868
  32. [32] Dungey N., Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z.247 (4) (2004) 765-794. Zbl1080.58022MR2077420
  33. [33] Dungey N., Riesz transforms on a discrete group of polynomial growth, Bull. London Math. Soc.36 (6) (2004) 833-840. Zbl1072.43002MR2083759
  34. [34] Dungey N., Some gradient estimates on covering manifolds, preprint. Zbl1112.58027MR2128280
  35. [35] Duong X.T., McIntosh A., Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamer.15 (2) (1999) 233-265. Zbl0980.42007MR1715407
  36. [36] Duong X.T., Robinson D., Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal.142 (1) (1996) 89-128. Zbl0932.47013MR1419418
  37. [37] Duong X.T., Yan L.X., New function spaces of BMO type, John–Nirenberg inequality, interpolation and applications, Comm. Pure Appl. Math., in press. Zbl1153.26305MR2162784
  38. [38] ter Elst A.F.M., Robinson D.W., Sikora A., Riesz transforms and Lie groups of polynomial growth, J. Funct. Anal.162 (1) (1999) 14-51. Zbl0953.22011MR1674538
  39. [39] Elworthy K., Li X.-M., Formulae for the derivatives of heat semigroups, J. Funct. Anal.125 (1) (1994) 252-286. Zbl0813.60049MR1297021
  40. [40] Feller W., An Introduction to Probability Theory and its Applications, vol. I, Wiley, New York, 1968. Zbl0039.13201MR228020
  41. [41] Fefferman C., Stein E., H p spaces in several variables, Acta Math.129 (1972) 137-193. Zbl0257.46078MR447953
  42. [42] Gaudry G., Sjögren P., Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group, Math. Z.232 (2) (1999) 241-256. Zbl0936.43006MR1718685
  43. [43] Grigor'yan A., On stochastically complete manifolds, DAN SSSR290 (3) (1986) 534-537, in Russian; English translation:, Soviet Math. Doklady34 (2) (1987) 310-313. Zbl0632.58041MR860324
  44. [44] Grigor'yan A., The heat equation on non-compact Riemannian manifolds, Matem. Sbornik182 (1) (1991) 55-87, in Russian; English translation:, Math. USSR Sb.72 (1) (1992) 47-77. Zbl0776.58035MR1098839
  45. [45] Grigor'yan A., Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal.127 (1995) 363-389. Zbl0842.58070MR1317722
  46. [46] Grigor'yan A., Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom.45 (1997) 33-52. Zbl0865.58042MR1443330
  47. [47] Grigor'yan A., Estimates of heat kernels on Riemannian manifolds, in: Davies B., Safarov Y. (Eds.), Spectral Theory and Geometry, London Math. Soc. Lecture Note Series, vol. 273, 1999, pp. 140-225. Zbl0985.58007MR1736868
  48. [48] Hajłasz P., Sobolev spaces on an arbitrary metric space, Pot. Anal.5 (1996) 403-415. Zbl0859.46022
  49. [49] Hajłasz P., Koskela P., Sobolev meets Poincaré, CRAS Paris320 (1995) 1211-1215. Zbl0837.46024
  50. [50] Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc.145 (2000) 688. Zbl0954.46022
  51. [51] Hebisch W., A multiplier theorem for Schrödinger operators, Coll. Math.60/61 (1990) 659-664. Zbl0779.35025MR1096404
  52. [52] Hebisch W., Steger T., Multipliers and singular integrals on exponential growth groups, Math. Z.245 (2003) 35-61. Zbl1035.43001MR2023952
  53. [53] Hofmann S., Martell J.M., L p bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat.47 (2) (2003) 497-515. Zbl1074.35031MR2006497
  54. [54] Ishiwata S., A Berry–Esseen type theorem on a nilpotent covering graph, Canad. J. Math., submitted for publication. Zbl1062.22018
  55. [55] Ishiwata S., Asymptotic behavior of a transition probability for a random walk on a nilpotent covering graph, in: Discrete Geometric Analysis, Contemp. Math., vol. 347, Amer. Math. Soc., Providence, RI, 2004, pp. 57-68. Zbl1061.22009MR2077030
  56. [56] Iwaniec T., The Gehring lemma, in: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 181-204. Zbl0888.30017MR1488451
  57. [57] Iwaniec T., Scott C., Stroffolini B., Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica Pura ed Applicata, IVCLXXVII (1999) 37-115. Zbl0963.58003MR1747627
  58. [58] Li H.-Q., La transformation de Riesz sur les variétés coniques, J. Funct. Anal.168 (1999) 145-238. Zbl0937.43004MR1717835
  59. [59] Li H.-Q., Estimations du noyau de la chaleur sur les variétés coniques et ses applications, Bull. Sci. Math.124 (5) (2000) 365-384. Zbl0977.58024MR1781554
  60. [60] Li H.-Q., Analyse sur les variétés cuspidales, Math. Ann.326 (2003) 625-647. Zbl1052.58029MR2003446
  61. [61] Li H.-Q., Lohoué N., Transformées de Riesz sur une classe de variétés à singularités coniques, J. Math. Pures Appl.82 (2003) 275-312. Zbl1070.58024MR1993284
  62. [62] Li J., Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal.97 (1991) 293-310. Zbl0724.58064MR1111183
  63. [63] Li P., Yau S.T., On the parabolic kernel of the Schrödinger operator, Acta Math.156 (1986) 153-201. Zbl0611.58045MR834612
  64. [64] Li X.D., Riesz transforms and Schrödinger operators on complete Riemannian manifolds with negative Ricci curvature, preprint. 
  65. [65] Lohoué N., Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal.61 (2) (1985) 164-201. Zbl0605.58051MR786621
  66. [66] Lohoué N., Estimation des projecteurs de de Rham–Hodge de certaines variétés riemanniennes non compactes, unpublished manuscript, 1984. 
  67. [67] Lohoué N., Inégalités de Sobolev pour les formes différentielles sur une variété riemannienne, CRAS Paris, série I301 (6) (1985) 277-280. Zbl0581.53029MR803217
  68. [68] Lohoué N., Transformées de Riesz et fonctions de Littlewood–Paley sur les groupes non moyennables, CRAS Paris, série I306 (1988) 327-330. Zbl0661.43002MR934611
  69. [69] Lohoué N., Mustapha S., Sur les transformées de Riesz sur les espaces homogènes des groupes de Lie semi-simples, Bull. Soc. Math. France128 (4) (2000) 485-495. Zbl0987.22004MR1815395
  70. [70] Lohoué N., Mustapha S., Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes, Canad. J. Math.50 (5) (1998) 1090-1104. Zbl0915.22006MR1650934
  71. [71] Marias M., Russ E., H 1 -boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds, Ark. Mat.41 (1) (2003) 115-132. Zbl1038.42016MR1971944
  72. [72] Martell J.M., Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math.161 (2) (2004) 113-145. Zbl1044.42019MR2033231
  73. [73] Meyer P.-A., Transformations de Riesz pour les lois gaussiennes, in: Séminaire de Probabilités XVIII, Lecture Notes, vol. 1059, Springer, Berlin, 1984, pp. 179-193. Zbl0543.60078MR770960
  74. [74] Meyer Y., Ondelettes et opérateurs, tome II, Hermann, Paris, 1990. Zbl0694.41037MR1085487
  75. [75] Meyers N.G., An L p estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa3 (17) (1963) 189-206. Zbl0127.31904MR159110
  76. [76] Picard J., Gradient estimates for some diffusion semigroups, Probab. Theory Related Fields122 (2002) 593-612. Zbl0995.60075MR1902192
  77. [77] Piquard F., Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow, Publ. Mat.48 (2) (2004) 309-333. Zbl1061.43013MR2091008
  78. [78] Pisier G., Riesz transforms: a simpler analytic proof of P.A. Meyer's inequality, in: Séminaire de Probabilités XXII, Lecture Notes in Math., vol. 1321, Springer, Berlin, 1988, pp. 485-501. Zbl0645.60061MR960544
  79. [79] Qian Z., Gradient estimates and heat kernel estimates, Proc. Royal Soc. Edinburgh125A (1995) 975-990. Zbl0863.58064MR1361628
  80. [80] Rumin M., Differential geometry on C-C spaces and application to the Novikov–Shubin numbers of nilpotent Lie groups, CRAS Paris, série I329 (11) (1999) 985-990. Zbl0982.53029MR1733906
  81. [81] Rumin M., Around heat decay on forms and relations of nilpotent Lie groups, in: Séminaire de théorie spectrale et géométrie de Grenoble, vol. 19, 2000–2001, pp. 123-164. Zbl1035.58021MR1909080
  82. [82] Russ E., Riesz transforms on graphs, Math. Scand.87 (1) (2000) 133-160. Zbl1008.60085MR1776969
  83. [83] Saloff-Coste L., Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat.28 (1990) 315-331. Zbl0715.43009MR1084020
  84. [84] Saloff-Coste L., A note on Poincaré, Sobolev and Harnack inequalities, Duke J. Math.65 (1992) 27-38, I.R.M.N. Zbl0769.58054MR1150597
  85. [85] Saloff-Coste L., Parabolic Harnack inequality for divergence form second order differential operators, Pot. Anal.4 (4) (1995) 429-467. Zbl0840.31006MR1354894
  86. [86] Scott C., L p theory of differential forms on manifolds, Trans. Amer. Math. Soc.347 (1995) 2075-2096. Zbl0849.58002MR1297538
  87. [87] Shen Z., Bounds of Riesz transforms on L p spaces for second order elliptic operators,Ann. Inst. Fourier, in press. Zbl1068.47058
  88. [88] Sikora A., Riesz transform, Gaussian bounds and the method of wave equation, Math. Z.247 (3) (2004) 643-662. Zbl1066.58014MR2114433
  89. [89] Stein E.M., Topics in harmonic analysis related to the Littlewood–Paley theory, Princeton UP, 1970. Zbl0193.10502MR252961
  90. [90] Stein E.M., Some results in harmonic analysis in R n for n , Bull. Amer. Math. Soc.9 (1983) 71-73. Zbl0515.42018MR699317
  91. [91] Stein E.M., Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton UP, 1993. Zbl0821.42001MR1232192
  92. [92] Strichartz R., Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal.52 (1983) 48-79. Zbl0515.58037MR705991
  93. [93] Strichartz R., L p contractive projections and the heat semigroup for differential forms, J. Funct. Anal.65 (1986) 348-357. Zbl0587.58044MR826432
  94. [94] Stroock D., Applications of Fefferman–Stein type interpolation to probability and analysis, Comm. Pure Appl. Math.XXVI (1973) 477-495. Zbl0267.60051MR341601
  95. [95] Stroock D., Turetsky J., Upper bounds on derivatives of the logarithm of the heat kernel, Comm. Anal. Geom.6 (4) (1998) 669-685. Zbl0928.58031MR1664888
  96. [96] Thalmaier A., Wang F.-Y., Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal.155 (1) (1998) 109-124. Zbl0914.58042MR1622800
  97. [97] Thalmaier A., Wang F.-Y., Derivative estimates of semigroups and Riesz transforms on vector bundles, Pot. Anal.20 (2) (2004) 105-123. Zbl1048.58023MR2032944
  98. [98] Varopoulos N., Analysis on Lie groups, J. Funct. Anal.76 (1988) 346-410. Zbl0634.22008MR924464
  99. [99] Varopoulos N., Random walks and Brownian motion on manifolds, in: Analisi Armonica, Spazi Simmetrici e Teoria della Probabilità, Symposia Math., vol. XXIX, 1987, pp. 97-109. Zbl0651.60013MR951181
  100. [100] Yau S.T., Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J.25 (1976) 659-670. Zbl0335.53041MR417452

Citations in EuDML Documents

  1. Satoshi Ishiwata, Discrete version of Dungey’s proof for the gradient heat kernel estimate on coverings
  2. Zhongwei Shen, Bounds of Riesz Transforms on L p Spaces for Second Order Elliptic Operators
  3. Stefano Meda, Alcuni aspetti dell'analisi su varietà riemanniane
  4. Peter Sjögren, Maria Vallarino, Boundedness from H 1 to L 1 of Riesz transforms on a Lie group of exponential growth
  5. Pascal Auscher, Thierry Coulhon, Riesz transform on manifolds and Poincaré inequalitie

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