Riesz transform on manifolds and heat kernel regularity

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

doi:10.1016/j.ansens.2004.10.003

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

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 37 (2004) no. 6, pp. 911-957.

MR Zbl | 7 citations dans Numdam

DOI : 10.1016/j.ansens.2004.10.003

@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 = {https://www.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  - https://www.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 https://www.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. https://www.numdam.org/articles/10.1016/j.ansens.2004.10.003/

Bibliographie
Cité par

[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. | MR | Zbl

[2] Auscher P., 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] 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. | MR | Zbl

[4] Auscher P., Tchamitchian P., 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] Bakry D., Transformations de Riesz pour les semi-groupes symétriques, Seconde partie: étude sous la condition $Γ_{2} \geq 0$ , in: Séminaire de Probabilités XIX, Lecture Notes, vol. 1123, Springer, Berlin, 1985, pp. 145-174. | Numdam | MR | Zbl

[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. | Numdam | MR | Zbl

[8] Bakry D., The Riesz transforms associated with second order differential operators, in: Seminar on Stochastic Processes, vol. 88, Birkhäuser, Basel, 1989. | MR | Zbl

[9] Blunck S., Kunstmann P., Calderón-Zygmund theory for non-integral operators and the $H^{\infty}$ functional calculus, Rev. Mat. Iberoamer. 19 (3) (2003) 919-942. | MR | Zbl

[10] Caffarelli L., Peral I., On $W^{1, p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21. | MR | Zbl

[11] Carron G., Formes harmoniques $L^{2}$ sur les variétés non-compactes, Rend. Mat. Appl. 7 (21) (2001), 1-4, 87-119. | MR | Zbl

[12] Chavel I., 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] Coifman R., Rochberg R., Another characterization of BMO, Proc. Amer. Math. Soc. 79 (2) (1980) 249-254. | MR | Zbl

[15] Coifman R., Weiss G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569-645. | MR | Zbl

[16] Coulhon T., Noyau de la chaleur et discrétisation d'une variété riemannienne, Israel J. Math. 80 (1992) 289-300. | MR | Zbl

[17] Coulhon T., Off-diagonal heat kernel lower bounds without Poincaré, J. London Math. Soc. 68 (3) (2003) 795-816. | MR | Zbl

[18] Coulhon T., Duong X.T., Riesz transforms for $1 \leq p \leq 2$ , Trans. Amer. Math. Soc. 351 (1999) 1151-1169. | MR | Zbl

[19] Coulhon T., Duong X.T., Riesz transforms for $p > 2$ , CRAS Paris, série I 332 (11) (2001) 975-980. | MR | Zbl

[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. | MR | Zbl

[21] Coulhon T., Grigor'Yan A., Pittet Ch., A geometric approach to on-diagonal heat kernel lower bounds on groups, Ann. Inst. Fourier 51 (6) (2001) 1763-1827. | Numdam | MR | Zbl

[22] Coulhon T., Li H.Q., 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] Coulhon T., Müller D., Zienkiewicz J., About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (2) (1996) 369-379. | MR | Zbl

[24] Coulhon T., Saloff-Coste L., 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] Cranston M., Gradient estimates on manifolds using coupling, J. Funct. Anal. 99 (1) (1991) 110-124. | MR | Zbl

[27] Davies E.B., Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math. 58 (1992) 99-119. | MR | Zbl

[28] Davies E.B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169. | MR | Zbl

[29] De Rham G., 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] Dungey N., Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (4) (2004) 765-794. | MR | Zbl

[33] Dungey N., 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] Duong X.T., Mcintosh A., Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamer. 15 (2) (1999) 233-265. | MR | Zbl

[36] Duong X.T., Robinson D., 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] 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. | MR | Zbl

[39] Elworthy K., Li X.-M., Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1) (1994) 252-286. | MR | Zbl

[40] Feller W., An Introduction to Probability Theory and its Applications, vol. I, Wiley, New York, 1968. | MR | Zbl

[41] Fefferman C., Stein E., $H^{p}$ spaces in several variables, Acta Math. 129 (1972) 137-193. | MR | Zbl

[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. | MR | Zbl

[43] Grigor'Yan A., 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] Grigor'Yan A., 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] Grigor'Yan A., Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995) 363-389. | MR | Zbl

[46] Grigor'Yan A., Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. 45 (1997) 33-52. | MR | Zbl

[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. | MR | Zbl

[48] Hajłasz P., Sobolev spaces on an arbitrary metric space, Pot. Anal. 5 (1996) 403-415. | Zbl

[49] Hajłasz P., Koskela P., Sobolev meets Poincaré, CRAS Paris 320 (1995) 1211-1215. | Zbl

[50] Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000) 688. | Zbl

[51] Hebisch W., A multiplier theorem for Schrödinger operators, Coll. Math. 60/61 (1990) 659-664. | MR | Zbl

[52] Hebisch W., Steger T., Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 35-61. | MR | Zbl

[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. | MR | Zbl

[54] Ishiwata S., A Berry-Esseen type theorem on a nilpotent covering graph, Canad. J. Math., submitted for publication. | Zbl

[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. | MR | Zbl

[56] Iwaniec T., The Gehring lemma, in: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 181-204. | MR | Zbl

[57] Iwaniec T., Scott C., Stroffolini B., Nonlinear Hodge theory on manifolds with boundary, Annali di Matematica Pura ed Applicata, IV CLXXVII (1999) 37-115. | MR | Zbl

[58] Li H.-Q., La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999) 145-238. | MR | Zbl

[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. | MR | Zbl

[60] Li H.-Q., Analyse sur les variétés cuspidales, Math. Ann. 326 (2003) 625-647. | MR | Zbl

[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. | MR | Zbl

[62] Li J., Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal. 97 (1991) 293-310. | MR | Zbl

[63] Li P., Yau S.T., 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] 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. | MR | Zbl

[66] Lohoué N., Estimation des projecteurs de de Rham-Hodge de certaines variétés riemanniennes non compactes, unpublished manuscript, 1984.

[67] Lohoué N., Inégalités de Sobolev pour les formes différentielles sur une variété riemannienne, CRAS Paris, série I 301 (6) (1985) 277-280. | MR | Zbl

[68] Lohoué N., Transformées de Riesz et fonctions de Littlewood-Paley sur les groupes non moyennables, CRAS Paris, série I 306 (1988) 327-330. | MR | Zbl

[69] Lohoué N., Mustapha S., 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] 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. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[74] Meyer Y., Ondelettes et opérateurs, tome II, Hermann, Paris, 1990. | MR | Zbl

[75] Meyers N.G., An $L^{p}$ 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] Picard J., Gradient estimates for some diffusion semigroups, Probab. Theory Related Fields 122 (2002) 593-612. | MR | Zbl

[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. | MR | Zbl

[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. | Numdam | MR | Zbl

[79] Qian Z., Gradient estimates and heat kernel estimates, Proc. Royal Soc. Edinburgh 125A (1995) 975-990. | MR | Zbl

[80] Rumin M., 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] 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. | Numdam | MR | Zbl

[82] Russ E., Riesz transforms on graphs, Math. Scand. 87 (1) (2000) 133-160. | MR | Zbl

[83] Saloff-Coste L., Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat. 28 (1990) 315-331. | MR | Zbl

[84] Saloff-Coste L., A note on Poincaré, Sobolev and Harnack inequalities, Duke J. Math. 65 (1992) 27-38, I.R.M.N. | MR | Zbl

[85] Saloff-Coste L., Parabolic Harnack inequality for divergence form second order differential operators, Pot. Anal. 4 (4) (1995) 429-467. | MR | Zbl

[86] Scott C., $L^{p}$ theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995) 2075-2096. | MR | Zbl

[87] Shen Z., Bounds of Riesz transforms on $L^{p}$ spaces for second order elliptic operators, Ann. Inst. Fourier, in press. | Numdam | Zbl

[88] Sikora A., Riesz transform, Gaussian bounds and the method of wave equation, Math. Z. 247 (3) (2004) 643-662. | MR | Zbl

[89] Stein E.M., Topics in harmonic analysis related to the Littlewood-Paley theory, Princeton UP, 1970. | MR | Zbl

[90] Stein E.M., Some results in harmonic analysis in $R^{n}$ for $n \to \infty$ , Bull. Amer. Math. Soc. 9 (1983) 71-73. | MR | Zbl

[91] Stein E.M., Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton UP, 1993. | MR | Zbl

[92] Strichartz R., Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983) 48-79. | MR | Zbl

[93] Strichartz R., $L^{p}$ contractive projections and the heat semigroup for differential forms, J. Funct. Anal. 65 (1986) 348-357. | MR | Zbl

[94] Stroock D., Applications of Fefferman-Stein type interpolation to probability and analysis, Comm. Pure Appl. Math. XXVI (1973) 477-495. | MR | Zbl

[95] Stroock D., Turetsky J., Upper bounds on derivatives of the logarithm of the heat kernel, Comm. Anal. Geom. 6 (4) (1998) 669-685. | MR | Zbl

[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. | MR | Zbl

[97] Thalmaier A., Wang F.-Y., Derivative estimates of semigroups and Riesz transforms on vector bundles, Pot. Anal. 20 (2) (2004) 105-123. | MR | Zbl

[98] Varopoulos N., Analysis on Lie groups, J. Funct. Anal. 76 (1988) 346-410. | MR | Zbl

[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. | MR | Zbl

[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. | MR | Zbl

Cité par Sources :