@article{ASENS_2004_4_37_6_911_0, author = {Auscher, Pascal and Coulhon, Thierry and Duong, Xuan Thinh and Hofmann, Steve}, title = {Riesz transform on manifolds and heat kernel regularity}, journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure}, pages = {911--957}, publisher = {Elsevier}, volume = {Ser. 4, 37}, number = {6}, year = {2004}, doi = {10.1016/j.ansens.2004.10.003}, mrnumber = {2119242}, zbl = {1086.58013}, language = {en}, url = {http://archive.numdam.org/articles/10.1016/j.ansens.2004.10.003/} }
TY - JOUR 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 SP - 911 EP - 957 VL - 37 IS - 6 PB - Elsevier UR - http://archive.numdam.org/articles/10.1016/j.ansens.2004.10.003/ DO - 10.1016/j.ansens.2004.10.003 LA - en ID - ASENS_2004_4_37_6_911_0 ER -
%0 Journal Article %A Auscher, Pascal %A Coulhon, Thierry %A Duong, Xuan Thinh %A Hofmann, Steve %T Riesz transform on manifolds and heat kernel regularity %J Annales scientifiques de l'École Normale Supérieure %D 2004 %P 911-957 %V 37 %N 6 %I Elsevier %U http://archive.numdam.org/articles/10.1016/j.ansens.2004.10.003/ %R 10.1016/j.ansens.2004.10.003 %G en %F ASENS_2004_4_37_6_911_0
Auscher, Pascal; Coulhon, Thierry; Duong, Xuan Thinh; Hofmann, Steve. Riesz transform on manifolds and heat kernel regularity. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 37 (2004) no. 6, pp. 911-957. doi : 10.1016/j.ansens.2004.10.003. http://archive.numdam.org/articles/10.1016/j.ansens.2004.10.003/
[1] 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. | MR | Zbl
,[2] Auscher P., On necessary and sufficient conditions for -estimates of Riesz transforms associated to elliptic operators on and related estimates, preprint 2004-04, Université de Paris-Sud, Mathématiques.
[3] The solution of the Kato square root problem for second order elliptic operators on , Annals of Math. 156 (2002) 633-654. | MR | Zbl
, , , , ,[4] Square Root Problem for Divergence Operators and Related Topics, Astérisque, vol. 249, 1998. | Numdam | MR | Zbl
, ,[5] Auscher P., Coulhon T., Riesz transforms on manifolds and Poincaré inequalities, preprint, 2004. | Numdam | MR
[6] Transformations de Riesz pour les semi-groupes symétriques, Seconde partie: étude sous la condition , in: Séminaire de Probabilités XIX, Lecture Notes, vol. 1123, Springer, Berlin, 1985, pp. 145-174. | Numdam | MR | Zbl
,[7] 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. | Numdam | MR | Zbl
,[8] The Riesz transforms associated with second order differential operators, in: Seminar on Stochastic Processes, vol. 88, Birkhäuser, Basel, 1989. | MR | Zbl
,[9] Calderón-Zygmund theory for non-integral operators and the functional calculus, Rev. Mat. Iberoamer. 19 (3) (2003) 919-942. | MR | Zbl
, ,[10] On estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21. | MR | Zbl
, ,[11] Formes harmoniques sur les variétés non-compactes, Rend. Mat. Appl. 7 (21) (2001), 1-4, 87-119. | MR | Zbl
,[12] Riemannian Geometry: A Modern Introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. | MR | Zbl
,[13] Chen J.-C., Heat kernels on positively curved manifolds and applications, Ph. D. thesis, Hanghzhou university, 1987.
[14] Another characterization of BMO, Proc. Amer. Math. Soc. 79 (2) (1980) 249-254. | MR | Zbl
, ,[15] Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569-645. | MR | Zbl
, ,[16] Noyau de la chaleur et discrétisation d'une variété riemannienne, Israel J. Math. 80 (1992) 289-300. | MR | Zbl
,[17] Off-diagonal heat kernel lower bounds without Poincaré, J. London Math. Soc. 68 (3) (2003) 795-816. | MR | Zbl
,[18] Riesz transforms for , Trans. Amer. Math. Soc. 351 (1999) 1151-1169. | MR | Zbl
, ,[19] Riesz transforms for , CRAS Paris, série I 332 (11) (2001) 975-980. | MR | Zbl
, ,[20] Riesz transform and related inequalities on non-compact Riemannian manifolds, Comm. Pure Appl. Math. 56 (12) (2003) 1728-1751. | MR | Zbl
, ,[21] A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier 51 (6) (2001) 1763-1827. | Numdam | MR | Zbl
, , ,[22] Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz, Archiv der Mathematik 83 (2004) 229-242. | MR | Zbl
, ,[23] About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (2) (1996) 369-379. | MR | Zbl
, , ,[24] Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamer. 9 (2) (1993) 293-314. | MR | Zbl
, ,[25] Coulhon T., Sikora A., Gaussian heat kernel bounds via Phragmén-Lindelöf theorems, preprint.
[26] Gradient estimates on manifolds using coupling, J. Funct. Anal. 99 (1) (1991) 110-124. | MR | Zbl
,[27] Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math. 58 (1992) 99-119. | MR | Zbl
,[28] Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169. | MR | Zbl
,[29] Variétés différentiables, formes, courants, formes harmoniques, Hermann, Paris, 1973. | MR | Zbl
,[30] Dragičević O., Volberg A., Bellman functions and dimensionless estimates of Riesz transforms, preprint.
[31] Driver B., Melcher T., Hypoelliptic heat kernel inequalities on the Heisenberg group, J. Funct. Anal., appeared online 11 September 2004. | MR | Zbl
[32] Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (4) (2004) 765-794. | MR | Zbl
,[33] Riesz transforms on a discrete group of polynomial growth, Bull. London Math. Soc. 36 (6) (2004) 833-840. | MR | Zbl
,[34] Dungey N., Some gradient estimates on covering manifolds, preprint. | MR
[35] Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamer. 15 (2) (1999) 233-265. | MR | Zbl
, ,[36] Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal. 142 (1) (1996) 89-128. | MR | Zbl
, ,[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. | MR | Zbl
[38] Riesz transforms and Lie groups of polynomial growth, J. Funct. Anal. 162 (1) (1999) 14-51. | MR | Zbl
, , ,[39] Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1) (1994) 252-286. | MR | Zbl
, ,[40] An Introduction to Probability Theory and its Applications, vol. I, Wiley, New York, 1968. | MR | Zbl
,[41] spaces in several variables, Acta Math. 129 (1972) 137-193. | MR | Zbl
, ,[42] Haar-like expansions and boundedness of a Riesz operator on a solvable Lie group, Math. Z. 232 (2) (1999) 241-256. | MR | Zbl
, ,[43] On stochastically complete manifolds, DAN SSSR 290 (3) (1986) 534-537, in Russian; English translation:, Soviet Math. Doklady 34 (2) (1987) 310-313. | MR | Zbl
,[44] The heat equation on non-compact Riemannian manifolds, Matem. Sbornik 182 (1) (1991) 55-87, in Russian; English translation:, Math. USSR Sb. 72 (1) (1992) 47-77. | MR | Zbl
,[45] Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995) 363-389. | MR | Zbl
,[46] Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. 45 (1997) 33-52. | MR | Zbl
,[47] Estimates of heat kernels on Riemannian manifolds, in: , (Eds.), Spectral Theory and Geometry, London Math. Soc. Lecture Note Series, vol. 273, 1999, pp. 140-225. | MR | Zbl
,[48] Sobolev spaces on an arbitrary metric space, Pot. Anal. 5 (1996) 403-415. | Zbl
,[49] Sobolev meets Poincaré, CRAS Paris 320 (1995) 1211-1215. | Zbl
, ,[50] Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000) 688. | Zbl
, ,[51] A multiplier theorem for Schrödinger operators, Coll. Math. 60/61 (1990) 659-664. | MR | Zbl
,[52] Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 35-61. | MR | Zbl
, ,[53] bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2) (2003) 497-515. | MR | Zbl
, ,[54] Ishiwata S., A Berry-Esseen type theorem on a nilpotent covering graph, Canad. J. Math., submitted for publication. | Zbl
[55] 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. | MR | Zbl
,[56] The Gehring lemma, in: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 181-204. | MR | Zbl
,[57] Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica Pura ed Applicata, IV CLXXVII (1999) 37-115. | MR | Zbl
, , ,[58] La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999) 145-238. | MR | Zbl
,[59] Estimations du noyau de la chaleur sur les variétés coniques et ses applications, Bull. Sci. Math. 124 (5) (2000) 365-384. | MR | Zbl
,[60] Analyse sur les variétés cuspidales, Math. Ann. 326 (2003) 625-647. | MR | Zbl
,[61] Transformées de Riesz sur une classe de variétés à singularités coniques, J. Math. Pures Appl. 82 (2003) 275-312. | MR | Zbl
, ,[62] Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal. 97 (1991) 293-310. | MR | Zbl
,[63] On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153-201. | MR | Zbl
, ,[64] Li X.D., Riesz transforms and Schrödinger operators on complete Riemannian manifolds with negative Ricci curvature, preprint.
[65] 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. | MR | Zbl
,[66] Lohoué N., Estimation des projecteurs de de Rham-Hodge de certaines variétés riemanniennes non compactes, unpublished manuscript, 1984.
[67] Inégalités de Sobolev pour les formes différentielles sur une variété riemannienne, CRAS Paris, série I 301 (6) (1985) 277-280. | MR | Zbl
,[68] Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables, CRAS Paris, série I 306 (1988) 327-330. | MR | Zbl
,[69] Sur les transformées de Riesz sur les espaces homogènes des groupes de Lie semi-simples, Bull. Soc. Math. France 128 (4) (2000) 485-495. | Numdam | MR | Zbl
, ,[70] Sur les transformées de Riesz sur les groupes de Lie moyennables et sur certains espaces homogènes, Canad. J. Math. 50 (5) (1998) 1090-1104. | MR | Zbl
, ,[71] -boundedness of Riesz transforms and imaginary powers of the Laplacian on Riemannian manifolds, Ark. Mat. 41 (1) (2003) 115-132. | MR | Zbl
, ,[72] Sharp maximal functions associated with approximations of the identity in spaces of homogeneous type and applications, Studia Math. 161 (2) (2004) 113-145. | MR | Zbl
,[73] Transformations de Riesz pour les lois gaussiennes, in: Séminaire de Probabilités XVIII, Lecture Notes, vol. 1059, Springer, Berlin, 1984, pp. 179-193. | Numdam | MR | Zbl
,[74] Ondelettes et opérateurs, tome II, Hermann, Paris, 1990. | MR | Zbl
,[75] An estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa 3 (17) (1963) 189-206. | Numdam | MR | Zbl
,[76] Gradient estimates for some diffusion semigroups, Probab. Theory Related Fields 122 (2002) 593-612. | MR | Zbl
,[77] Riesz transforms on generalized Heisenberg groups and Riesz transforms associated to the CCR heat flow, Publ. Mat. 48 (2) (2004) 309-333. | MR | Zbl
,[78] 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. | Numdam | MR | Zbl
,[79] Gradient estimates and heat kernel estimates, Proc. Royal Soc. Edinburgh 125A (1995) 975-990. | MR | Zbl
,[80] Differential geometry on C-C spaces and application to the Novikov-Shubin numbers of nilpotent Lie groups, CRAS Paris, série I 329 (11) (1999) 985-990. | MR | Zbl
,[81] 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. | Numdam | MR | Zbl
,[82] Riesz transforms on graphs, Math. Scand. 87 (1) (2000) 133-160. | MR | Zbl
,[83] Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat. 28 (1990) 315-331. | MR | Zbl
,[84] A note on Poincaré, Sobolev and Harnack inequalities, Duke J. Math. 65 (1992) 27-38, I.R.M.N. | MR | Zbl
,[85] Parabolic Harnack inequality for divergence form second order differential operators, Pot. Anal. 4 (4) (1995) 429-467. | MR | Zbl
,[86] theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995) 2075-2096. | MR | Zbl
,[87] Shen Z., Bounds of Riesz transforms on spaces for second order elliptic operators, Ann. Inst. Fourier, in press. | Numdam | Zbl
[88] Riesz transform, Gaussian bounds and the method of wave equation, Math. Z. 247 (3) (2004) 643-662. | MR | Zbl
,[89] Topics in harmonic analysis related to the Littlewood-Paley theory, Princeton UP, 1970. | MR | Zbl
,[90] Some results in harmonic analysis in for , Bull. Amer. Math. Soc. 9 (1983) 71-73. | MR | Zbl
,[91] Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton UP, 1993. | MR | Zbl
,[92] Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983) 48-79. | MR | Zbl
,[93] contractive projections and the heat semigroup for differential forms, J. Funct. Anal. 65 (1986) 348-357. | MR | Zbl
,[94] Applications of Fefferman-Stein type interpolation to probability and analysis, Comm. Pure Appl. Math. XXVI (1973) 477-495. | MR | Zbl
,[95] Upper bounds on derivatives of the logarithm of the heat kernel, Comm. Anal. Geom. 6 (4) (1998) 669-685. | MR | Zbl
, ,[96] Gradient estimates for harmonic functions on regular domains in Riemannian manifolds, J. Funct. Anal. 155 (1) (1998) 109-124. | MR | Zbl
, ,[97] Derivative estimates of semigroups and Riesz transforms on vector bundles, Pot. Anal. 20 (2) (2004) 105-123. | MR | Zbl
, ,[98] Analysis on Lie groups, J. Funct. Anal. 76 (1988) 346-410. | MR | Zbl
,[99] Random walks and Brownian motion on manifolds, in: Analisi Armonica, Spazi Simmetrici e Teoria della Probabilità, Symposia Math., vol. XXIX, 1987, pp. 97-109. | MR | Zbl
,[100] Some function-theoretic properties of complete Riemannian manifolds and their applications to geometry, Indiana Univ. Math. J. 25 (1976) 659-670. | MR | Zbl
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