Riesz transform on manifolds and heat kernel regularity

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

Annales scientifiques de l'École Normale Supérieure, Tome 37 (2004), p. 911-957 / Harvested from Numdam

Détail

Publié le : 2004-01-01

MR 2119242 | Zbl 1086.58013 | 2 citations dans Numdam

DOI : https://doi.org/10.1016/j.ansens.2004.10.003

Bibliographie

[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 1178564 | Zbl 0792.22005

[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 1933726 | Zbl 01851146

[4] Auscher P., Tchamitchian P., Square Root Problem for Divergence Operators and Related Topics, Astérisque, vol. 249, 1998. | MR 1651262 | Zbl 0909.35001

[5] Auscher P., Coulhon T., Riesz transforms on manifolds and Poincaré inequalities, preprint, 2004. | Numdam | MR 2185868

[6] Bakry D., Transformations de Riesz pour les semi-groupes symétriques, Seconde partie: étude sous la condition ${\Gamma }_{2}\ge 0$, in: Séminaire de Probabilités XIX, Lecture Notes, vol. 1123, Springer, Berlin, 1985, pp. 145-174. | Numdam | MR 889473 | Zbl 0561.42011

[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 941980 | Zbl 0629.58018

[8] Bakry D., The Riesz transforms associated with second order differential operators, in: Seminar on Stochastic Processes, vol. 88, Birkhäuser, Basel, 1989. | MR 990472 | Zbl 0689.58032

[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 2053568 | Zbl 1057.42010

[10] Caffarelli L., Peral I., On ${W}^{1\text{,}p}$ estimates for elliptic equations in divergence form, Comm. Pure Appl. Math. 51 (1998) 1-21. | MR 1486629 | Zbl 0906.35030

[11] Carron G., Formes harmoniques ${L}^{2}$ sur les variétés non-compactes, Rend. Mat. Appl. 7 (21) (2001), 1-4, 87-119. | MR 1884938 | Zbl 1049.58006

[12] Chavel I., Riemannian Geometry: A Modern Introduction, Cambridge Tracts in Mathematics, vol. 108, Cambridge University Press, Cambridge, 1993. | MR 1271141 | Zbl 0810.53001

[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 565349 | Zbl 0432.42016

[15] Coifman R., Weiss G., Extensions of Hardy spaces and their use in analysis, Bull. Amer. Math. Soc. 83 (1977) 569-645. | MR 447954 | Zbl 0358.30023

[16] Coulhon T., Noyau de la chaleur et discrétisation d'une variété riemannienne, Israel J. Math. 80 (1992) 289-300. | MR 1202573 | Zbl 0772.58055

[17] Coulhon T., Off-diagonal heat kernel lower bounds without Poincaré, J. London Math. Soc. 68 (3) (2003) 795-816. | MR 2010012 | Zbl 1083.58025

[18] Coulhon T., Duong X.T., Riesz transforms for $1\le p\le 2$, Trans. Amer. Math. Soc. 351 (1999) 1151-1169. | MR 1458299 | Zbl 0973.58018

[19] Coulhon T., Duong X.T., Riesz transforms for $p>2$, CRAS Paris, série I 332 (11) (2001) 975-980. | MR 1838122 | Zbl 0987.43001

[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 2001444 | Zbl 1037.58017

[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 1871289 | Zbl 01710118

[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 2108551 | Zbl 1076.58017

[23] Coulhon T., Müller D., Zienkiewicz J., About Riesz transforms on the Heisenberg groups, Math. Ann. 305 (2) (1996) 369-379. | MR 1391221 | Zbl 0859.22006

[24] Coulhon T., Saloff-Coste L., Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamer. 9 (2) (1993) 293-314. | MR 1232845 | Zbl 0782.53066

[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 1120916 | Zbl 0770.58038

[27] Davies E.B., Heat kernel bounds, conservation of probability and the Feller property, J. d'Analyse Math. 58 (1992) 99-119. | MR 1226938 | Zbl 0808.58041

[28] Davies E.B., Uniformly elliptic operators with measurable coefficients, J. Funct. Anal. 132 (1995) 141-169. | MR 1346221 | Zbl 0839.35034

[29] De Rham G., Variétés différentiables, formes, courants, formes harmoniques, Hermann, Paris, 1973. | MR 346830 | Zbl 0065.32401

[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 2124868 | Zbl 1071.22005

[32] Dungey N., Heat kernel estimates and Riesz transforms on some Riemannian covering manifolds, Math. Z. 247 (4) (2004) 765-794. | MR 2077420 | Zbl 1080.58022

[33] Dungey N., Riesz transforms on a discrete group of polynomial growth, Bull. London Math. Soc. 36 (6) (2004) 833-840. | MR 2083759 | Zbl 1072.43002

[34] Dungey N., Some gradient estimates on covering manifolds, preprint. | MR 2128280

[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 1715407 | Zbl 0980.42007

[36] Duong X.T., Robinson D., Semigroup kernels, Poisson bounds and holomorphic functional calculus, J. Funct. Anal. 142 (1) (1996) 89-128. | MR 1419418 | Zbl 0932.47013

[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 2162784 | Zbl 02212448

[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 1674538 | Zbl 0953.22011

[39] Elworthy K., Li X.-M., Formulae for the derivatives of heat semigroups, J. Funct. Anal. 125 (1) (1994) 252-286. | MR 1297021 | Zbl 0813.60049

[40] Feller W., An Introduction to Probability Theory and its Applications, vol. I, Wiley, New York, 1968. | MR 228020 | Zbl 0039.13201

[41] Fefferman C., Stein E., ${H}^{p}$ spaces in several variables, Acta Math. 129 (1972) 137-193. | MR 447953 | Zbl 0257.46078

[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 1718685 | Zbl 0936.43006

[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 860324 | Zbl 0632.58041

[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 1098839 | Zbl 0776.58035

[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 1317722 | Zbl 0842.58070

[46] Grigor'Yan A., Gaussian upper bounds for the heat kernel on arbitrary manifolds, J. Diff. Geom. 45 (1997) 33-52. | MR 1443330 | Zbl 0865.58042

[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 1736868 | Zbl 0985.58007

[48] Hajłasz P., Sobolev spaces on an arbitrary metric space, Pot. Anal. 5 (1996) 403-415. | Zbl 0859.46022

[49] Hajłasz P., Koskela P., Sobolev meets Poincaré, CRAS Paris 320 (1995) 1211-1215. | Zbl 0837.46024

[50] Hajłasz P., Koskela P., Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000) 688. | Zbl 0954.46022

[51] Hebisch W., A multiplier theorem for Schrödinger operators, Coll. Math. 60/61 (1990) 659-664. | MR 1096404 | Zbl 0779.35025

[52] Hebisch W., Steger T., Multipliers and singular integrals on exponential growth groups, Math. Z. 245 (2003) 35-61. | MR 2023952 | Zbl 1035.43001

[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 2006497 | Zbl 1074.35031

[54] Ishiwata S., A Berry-Esseen type theorem on a nilpotent covering graph, Canad. J. Math., submitted for publication. | Zbl 1062.22018

[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 2077030 | Zbl 1061.22009

[56] Iwaniec T., The Gehring lemma, in: Quasiconformal Mappings and Analysis (Ann Arbor, MI, 1995), Springer, New York, 1998, pp. 181-204. | MR 1488451 | Zbl 0888.30017

[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 1747627 | Zbl 0963.58003

[58] Li H.-Q., La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999) 145-238. | MR 1717835 | Zbl 0937.43004

[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 1781554 | Zbl 0977.58024

[60] Li H.-Q., Analyse sur les variétés cuspidales, Math. Ann. 326 (2003) 625-647. | MR 2003446 | Zbl 1052.58029

[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 1993284 | Zbl 1070.58024

[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 1111183 | Zbl 0724.58064

[63] Li P., Yau S.T., On the parabolic kernel of the Schrödinger operator, Acta Math. 156 (1986) 153-201. | MR 834612 | Zbl 0611.58045

[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 786621 | Zbl 0605.58051

[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 803217 | Zbl 0581.53029

[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 934611 | Zbl 0661.43002

[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 1815395 | Zbl 0987.22004

[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 1650934 | Zbl 0915.22006

[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 1971944 | Zbl 1038.42016

[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 2033231 | Zbl 1044.42019

[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 770960 | Zbl 0543.60078

[74] Meyer Y., Ondelettes et opérateurs, tome II, Hermann, Paris, 1990. | MR 1085487 | Zbl 0694.41037

[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 159110 | Zbl 0127.31904

[76] Picard J., Gradient estimates for some diffusion semigroups, Probab. Theory Related Fields 122 (2002) 593-612. | MR 1902192 | Zbl 0995.60075

[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 2091008 | Zbl 1061.43013

[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 960544 | Zbl 0645.60061

[79] Qian Z., Gradient estimates and heat kernel estimates, Proc. Royal Soc. Edinburgh 125A (1995) 975-990. | MR 1361628 | Zbl 0863.58064

[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 1733906 | Zbl 0982.53029

[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 1909080 | Zbl 1035.58021

[82] Russ E., Riesz transforms on graphs, Math. Scand. 87 (1) (2000) 133-160. | MR 1776969 | Zbl 1008.60085

[83] Saloff-Coste L., Analyse sur les groupes de Lie à croissance polynomiale, Ark. Mat. 28 (1990) 315-331. | MR 1084020 | Zbl 0715.43009

[84] Saloff-Coste L., A note on Poincaré, Sobolev and Harnack inequalities, Duke J. Math. 65 (1992) 27-38, I.R.M.N. | MR 1150597 | Zbl 0769.58054

[85] Saloff-Coste L., Parabolic Harnack inequality for divergence form second order differential operators, Pot. Anal. 4 (4) (1995) 429-467. | MR 1354894 | Zbl 0840.31006

[86] Scott C., ${L}^{p}$ theory of differential forms on manifolds, Trans. Amer. Math. Soc. 347 (1995) 2075-2096. | MR 1297538 | Zbl 0849.58002

[87] Shen Z., Bounds of Riesz transforms on ${L}^{p}$ spaces for second order elliptic operators, Ann. Inst. Fourier, in press. | Numdam | Zbl 1068.47058

[88] Sikora A., Riesz transform, Gaussian bounds and the method of wave equation, Math. Z. 247 (3) (2004) 643-662. | MR 2114433 | Zbl 1066.58014

[89] Stein E.M., Topics in harmonic analysis related to the Littlewood-Paley theory, Princeton UP, 1970. | MR 252961 | Zbl 0193.10502

[90] Stein E.M., Some results in harmonic analysis in ${R}^{n}$ for $n\rightarrow \infty $, Bull. Amer. Math. Soc. 9 (1983) 71-73. | MR 699317 | Zbl 0515.42018

[91] Stein E.M., Harmonic Analysis, Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton UP, 1993. | MR 1232192 | Zbl 0821.42001

[92] Strichartz R., Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal. 52 (1983) 48-79. | MR 705991 | Zbl 0515.58037

[93] Strichartz R., ${L}^{p}$ contractive projections and the heat semigroup for differential forms, J. Funct. Anal. 65 (1986) 348-357. | MR 826432 | Zbl 0587.58044

[94] Stroock D., Applications of Fefferman-Stein type interpolation to probability and analysis, Comm. Pure Appl. Math. XXVI (1973) 477-495. | MR 341601 | Zbl 0267.60051

[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 1664888 | Zbl 0928.58031

[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 1622800 | Zbl 0914.58042

[97] Thalmaier A., Wang F.-Y., Derivative estimates of semigroups and Riesz transforms on vector bundles, Pot. Anal. 20 (2) (2004) 105-123. | MR 2032944 | Zbl 1048.58023

[98] Varopoulos N., Analysis on Lie groups, J. Funct. Anal. 76 (1988) 346-410. | MR 924464 | Zbl 0634.22008

[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 951181 | Zbl 0651.60013

[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 417452 | Zbl 0335.53041