Riesz transforms on forms and \(L^p\)-Hodge decomposition on complete Riemannian manifolds. (English) Zbl 1197.53052

Li, Xiang-Dong

Rev. Mat. Iberoam. 26, No. 2, 481-528 (2010); erratum ibid. 30, No. 1, 369-370 (2014).

Author’s abstract: “We prove the strong \(L^p\) stability of the heat semigroup generated by the Hodge Laplacian on complete Riemannian manifolds with non-negative Weitzenböck curvature. Based on a probabilistic representation formula, we obtain an explicit upper bound of the \(L^p\)-norm of the Riesz transforms on forms on complete Riemannian manifolds with suitable curvature conditions. Moreover, we establish the weak \(L^p\)-Hodge decomposition theorem on complete Riemannian manifolds with non-negative Weitzenböck curvature.”
Martingale representation of Riesz transforms in terms of covariant Ito calculus is done.

Reviewer: Simon Eloshvili (Tbilisi)

MSC:

53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions
58J65	Diffusion processes and stochastic analysis on manifolds
58J40	Pseudodifferential and Fourier integral operators on manifolds
60J65	Brownian motion

Keywords:

Hodge decomposition; martingale transforms; Riesz transforms; Weitzenböck curvature

References:

[1]	Arcozzi, N.: Riesz transforms on compact Lie groups, spheres and Gauss space. Ark. Mat. 36 (1998), no. 2, 201-231. · Zbl 1031.42009
[2]	Auscher, P., Coulhon, T., Duong, X.T. and Hofmann, S.: Riesz transform on manifolds and heat kernel regularity. Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 6, 911-957. · Zbl 1086.58013
[3]	Bakry, D.: Transformations de Riesz pour les semi-groupes symétriques. II. Étude sous la condition \(\Gamma_2\geq 0\). In Séminaire de Probabilités, XIX , 145-174. Lecture Notes in Math. 1123 . Springer, Berlin, 1985. · Zbl 0561.42011
[4]	Bakry, D.: Un critére de non-explosion pour certaines diffusions sur une variété Riemannienne compléte. C. R. Acad. Sci. Paris Sér. I. Math. 303 (1986), no. 1, 23-26. · Zbl 0589.60069
[5]	Bakry, D.: Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. In Séminaire de Probabilités, XXI , 137-172. Lecture Notes in Math. 1247 . Springer, Berlin, 1987. · Zbl 0629.58018
[6]	Bakry, D.: The Riesz transforms associated with second order differential operators. In Seminar on Stochastic Processes, 1988 , 1-43. Progr. Probab. 17 . Birkhäuser Boston, Boston, Ma, 1989. · Zbl 0689.58032
[7]	Berline, N., Getzler, E. and Vergne, M.: Heat kernels and Dirac operators. Grundlehren der Mathematischen Wissenschaften 298 . Springer-Verlag, Berlin, 1992. · Zbl 0744.58001
[8]	Bismut, J.M.: The Atiyah-Singer theorems: a probabilistic approach. I. The index theorem. J. Funct. Anal. 57 (1984), no. 1, 56-99. · Zbl 0538.58033
[9]	Bañuelos, R. and Lindeman, A.II.: A martingale study of the Beurling-Ahlfors transforms in \(\mathbbR^n\). J. Funct. Anal. 145 (1997), no. 1, 224-265. · Zbl 0876.60026
[10]	Bañuelos, R. and Wang, G.: Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80 (1995), no. 3, 575-600. · Zbl 0853.60040
[11]	Bañuelos, R. and Méndez-Hernández, P.: Space-time Brownian motion and the Beurling-Ahlfors transform. Indiana Univ. Math J. 52 (2003), no. 4, 981-990. · Zbl 1080.60043
[12]	Bueler, E.L.: The heat kernel weighted Hodge laplacian on noncompact manifolds. Trans. Amer. Math. Soc. 351 (1999), no. 2, 683-713. JSTOR: · Zbl 0920.58002
[13]	Burkholder, D.L.: A sharp and strict \(L^p\)-inequality for stochastic integrals. Ann. Probab. 15 (1987), no. 1, 268-273. · Zbl 0617.60042
[14]	Carron, G., Coulhon, T. and Hassell, A.: Riesz transform and \(L^p\) cohomology for manifolds with Euclidean ends. Duke Math. J. 133 (2006), no. 1, 59-93. · Zbl 1106.58021
[15]	Chernoff, P.R.: Essential self-adjointness of powers of generators of hyperbolic equations. J. Funct. Anal. 12 (1973), 401-414. · Zbl 0263.35066
[16]	Coulhon, T. and Duong, X.T.: Riesz transforms for \(1\leq p\leq 2\). Trans. Amer. Math. Soc. 351 (1999), no. 3, 1151-1169. JSTOR: · Zbl 0973.58018
[17]	Coulhon, T. and Duong, X.T.: Riesz transform and related inequalities on noncompact Riemannian manifolds. Comm. Pure Appl. Math. 56 (2003), no. 12, 1728-1751. · Zbl 1037.58017
[18]	Dellacherie, C., Maisonneuve, B. and Meyer, P.A.: Probabilies et Potentiel, Chapitres XVII a XXIV, Processus de Markov (fin) Complements de Calcul Stochastiques . Hermann, 1992.
[19]	De Rham, G.: Variétés Différentielles. Hermain, Paris, 1960.
[20]	Donaldson, S. and Sullivan, D.: Quasiconformal 4-manifolds. Acta Math. 163 (1989), no. 3-4, 181-252. · Zbl 0704.57008
[21]	Elworthy, D., LeJan, Y. and Li, X.M.: On the geometry of diffusion operators and stochastic flows. Lecture Notes in Mathematics 1720 . Springer-Verlag, Berlin, 1999. · Zbl 0942.58004
[22]	Engel, K.J. and Nagel, R.: One-parameter semigroups for linear evolution equations. Graduate Texts in Mathematics 194 . Springer-Verlag, New York, 2000. · Zbl 0952.47036
[23]	Gaffney, M.: A special Stokes’s theorem for complete Riemannian manifolds. Ann. of Math. (2) 60 (1954), 140-145. JSTOR: · Zbl 0055.40301
[24]	Gaffney, M.: Hilbert spaces methodes in the theory of harmonic integrals. Trans. Amer. Math. Soc. 78 (1955), 426-444. JSTOR: · Zbl 0064.34303
[25]	Gallot, S. and Meyer, D.: Opérateur de courbure et laplacien des formes différentielles d’une variété riemannnienne. J. Math. Pures Appl. (9) 54 (1975), no. 3, 259-284. · Zbl 0316.53036
[26]	Gilbarg, D. and Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Math. Springer-Verlag, Berlin, 2001. · Zbl 1042.35002
[27]	Grigor’yan, A. A.: On stochastically complete manifolds. DAN SSSR 290 (1986), 534-537. Engl. transl. Soviet Math. Dokl. 34 (1987), 310-313. · Zbl 0632.58041
[28]	Grigor’yan, A. A.: Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds. Bull. Amer. Math. Soc. (N. S.) 36 (1999), no. 2, 135-249. · Zbl 0927.58019
[29]	Gundy, R.F.: Sur les transformations de Riesz pour le semi-groupe d’Ornstein-Uhlenbeck. C. R. Acad. Sci. Paris Sér. I. Math 303 (1986), no. 19, 967-970. · Zbl 0606.60063
[30]	Gundy, R.F.: Some Topics in Probability and Analysis. CBMS Regional Conference Series in Math. 70 . Amer. Math. Soc., Providence, RI, 1989. · Zbl 0673.60050
[31]	Gundy, R.F. and Varopoulos, N.Th.: Les transformations de Riesz et les intégrales stochastiques. C.R. Acad. Sci. Paris Sér. A-B. 289 (1979), no. 1, A13-A16. · Zbl 0413.60003
[32]	Helmholtz, H.: Über Integrale der hydrodynamischen Gle ichungen, welche den Wirbelbewegunggen entsprechen. J. Reine Angew. Math. 55 (1858), 25-55. · ERAM 055.1448cj
[33]	Iwaniec, T. and Martin, G.: Quasiregular mappings in even dimensions. Acta Math. 170 (1993), no. 1, 29-81. · Zbl 0785.30008
[34]	Iwaniec, T. and Martin, G.: Riesz transforms and related singular integrals. J. Reine Angew. Math. 473 (1996), 25-57. · Zbl 0847.42015
[35]	Jacob, N.: Pseudo differential operators and Markov processes. Vol. I. Fourier analysis and semigroups. Imperial College Press, London, 2002.
[36]	Karp, L. and Li, P.: The heat equation on complete Riemannian manifolds. Unpublished manuscript, 1982.
[37]	Larsso-Cohn, L.: On the constants in the Meyer inequality. Monatsch. Math. 137 (2002), no. 1, 51-56. · Zbl 1005.60069
[38]	Leray, J.: Sur le mouvement d’un liquide visqueux emplissant l’espace. Acta Math. 63 (1934), no. 1, 193-248. · JFM 60.0726.05
[39]	Li, P. and Schoen, R.: \(L^p\) and mean value properties of subharmonic functions on Riemannian manifolds. Acta Math. 153 (1984), no. 3-4, 279-301. · Zbl 0556.31005
[40]	Li, X.D.: Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pures Appl. (9) 84 (2005), no. 10, 1295-1361. · Zbl 1082.58036
[41]	Li, X.D.: Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. Rev. Mat. Iberoamericana 22 (2006), no. 2, 591-648. · Zbl 1119.53022
[42]	Li, X.D.: Martingale transforms and \(L^p\)-norm estimates of Riesz transforms on complete Riemannian manifolds. Probab. Theory Related Fields 141 (2008), no. 1-2, 247-281. · Zbl 1140.53022
[43]	Li, X.D.: Sobolev inequalities on differential forms and \(L^p, q\)-cohomology on complete Riemannian manifolds. J. Geom. Anal. 20 (2010), 354-387. · Zbl 1192.46031
[44]	Li, X.D.: On the weak \(L^p\)-Hodge decomposition over complete Riemannian manifolds. Probab. Theory Related Fields , doi: 10.1007/s00440-010-0270-2.
[45]	Li, X.D.: On the strong \(L^p\)-Hodge decomposition over complete Riemannian manifolds. J. Funct. Anal. 257 (2009), no. 11, 3617-3646. · Zbl 1196.58001
[46]	Li, X.D.: \(L^p\)-estimates and existence theorems of the \(\bar\partial\)-operator on complete Kähler manifolds. Adv. in Math. 224 (2010), 620-647. · Zbl 1208.53073
[47]	Lions, P.L.: Mathematical Topics in Fluid Mechanics, Vol 1, Incompressible Methods, and Vol 2 Compressible Models. Oxford Lecture Series in Mathematics and its Applications 3 and 10 . The Clarendon Press, Oxford University Press, New York, 1966. · Zbl 0866.76002
[48]	Lohoué, N.: Comparaison des champs de vecteurs et des puissances du Laplacien sur une variété riemannienne à courbure non positive. J. Funct. Anal. 61 (1985), no. 2, 164-201. · Zbl 0605.58051
[49]	Lohoué, N.: Estimation des projecteurs de de Rham Hodge de certaines variétés Riemanniennes non-compactes. Math. Nachr. 279 (2006), no. 3, 272-298. · Zbl 1095.58006
[50]	Malliavin, P.: Formule de la moyenne, calcul des perturbations et théorèmes d’annulation pour les formes harmoniques. J. Functional Analysis 17 (1974), 274-291. · Zbl 0425.58022
[51]	Meyer, P.A.: Démonstrations probabilistes des inégalités de Littlewood-Paley. I. Les inégalités classiques. In Séminaire de Probabilités, X , 125-141. Lecture Notes in Math. 511 . Springer, Berlin, 1976. · Zbl 0332.60032
[52]	Meyer, P.A.: Le dual de \(H^1(\mathbbR^v)\): démonstrations probabilistes. In Séminaire de Probabilités, XI , 132-195. Lecture Notes in Math. 581 . Springer, Berlin, 1977. · Zbl 0371.60057
[53]	Meyer, P.A.: Transformations de Riesz pour les lois gaussiennes. In Seminar on Probability, XVIII , 179-193. Lecture Notes in Math. 1059 . Springer, Berlin, 1984. · Zbl 0543.60078
[54]	Milgram, A. and Rosenbloom, P.: Harmonic forms and heat conduction, I: Closed Riemannian manifolds. Proc. Nat. Acad. Sci. 37 (1951), 180-184. JSTOR: · Zbl 0044.31703
[55]	Morrey, C.B.Jr.: Multiple integrals in the calculus of variations. Die Grundlehren der mathematischen Wissenschaften 130 . Springer-Verlag, New York, 1966. · Zbl 0142.38701
[56]	Norris, J.R.: A complete differential formalism for stochastic calculus in manifolds. In Séminaire de Probabilités XXVI , 189-209. Lecture Notes in Math. 1526 . Springer, Berlin, 1992. · Zbl 0791.58111
[57]	Pichorides, S.K.: On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov. Studia Math. 44 (1972), 165-179. · Zbl 0238.42007
[58]	Robbin, J.W., Rogers, R.C. and Temple, B.: On weak continuity and the Hodge decomposition. Trans. Amer. Math. Soc. 303 (1987), no. 2, 609-618. JSTOR: · Zbl 0634.35005
[59]	Rogers, L.C.G. and Williams, D.: Diffusions, Markov processes and martingales, Vol. 1. Foundations. Cambridge Mathematical Library. Cambridge University Press, Cambridge, 2000. · Zbl 0977.60005
[60]	Scott, C.: \(L^p\) theory of differential forms on manifolds. Trans. Amer. Math. Soc. 347 (1995), no. 6, 2075-2096. JSTOR: · Zbl 0849.58002
[61]	Shigekawa, I.: Stochastic Analysis. Translations of Mathematical Monographs 224 . Iwanami Series in Modern Mathematics. American Mathematical Society, Providence, RI, 2004. · Zbl 1064.60003
[62]	Shigekawa, I. and Yoshida, N.: Littlewood-Paley-Stein inequality for a symmetric diffusion. J. Math. Soc. Japan 44 (1992), no. 2, 251-280. · Zbl 0764.60060
[63]	Stein, E.M.: Singular integrals and differentiability properties of functions. Princeton Mathematical Series 30 . Princeton University Press, Princeton, N.J., 1970. · Zbl 0207.13501
[64]	Stein, E.M.: Topics in harmonic analysis related to the Littlewood-Paley theory. Annals of Mathematics Studies 63 . Princeton University Press, Princeton, N.J.; University Tokyo Press, Tokyo, 1970. · Zbl 0193.10502
[65]	Strichartz, R.: Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52 (1983), no. 1, 48-79. · Zbl 0515.58037
[66]	Strichartz, R.: \(L^p\)-contractive projections and the heat semigroup for differential forms. J. Funct. Anal. 65 (1986), no. 3, 348-357. · Zbl 0587.58044
[67]	Stroock, D.W. and Varadhan, S.R.S.: Multidimensional Diffusion Processes. Grundlehren der Mathematischen Wissenschaften 233 . Springer-Verlag, Berlin-New York, 1979. · Zbl 0426.60069
[68]	Sturm, K.T.: Analysis on local Dirichlet spaces. I. Recurrence, conservativeness and \(L^p\)-Liouville properties. J. Reine Angew. Math. 456 (1994), 173-196. · Zbl 0806.53041
[69]	Witten, E.: Supersymmetry and Morse theory. J. Differential Geom. 17 (1982), no. 4, 661-692. · Zbl 0499.53056
[70]	Yau, S.T.: On the heat kernel of a complete Riemannian manifold. J. Math. Pures Appl. (9) 57 (1978), no. 2, 191-201. · Zbl 0405.35025
[71]	Yoshida, N.: Sobolev spaces on a Riemannian manifold and their equivalence. J. Math. Kyoto Univ. 32 (1992), no. 3, 621-654. · Zbl 0771.58005
[72]	Yoshida, N.: The Littlewood-Paley-Stein inequality on an infinite dimensional manifold. J. Funct. Anal. 122 (1994), no. 2, 402-427. · Zbl 0804.58062