Abstract
Under the condition that the Bakry–Emery Ricci curvature is bounded from below, we prove a probabilistic representation formula of the Riesz transforms associated with a symmetric diffusion operator on a complete Riemannian manifold. Using the Burkholder sharp L p-inequality for martingale transforms, we obtain an explicit and dimension-free upper bound of the L p-norm of the Riesz transforms on such complete Riemannian manifolds for all 1 < p < ∞. In the Euclidean and the Gaussian cases, our upper bound is asymptotically sharp when p→ 1 and when p→ ∞.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
Auscher P., Coulhon T., Duong X.T. and Hofmann S. (2004). Riesz transform on manifolds and heat kernel regularity. Ann. Sci. Ecole Norm. Sup. 37(6): 911–957
Arcozzi N. (1998). Riesz transforms on compact Lie groups, spheres and Gauss space. Ark. Mat. 36: 201–231
Bakry D. (1986). Un critère de non-explosion pour certaines diffusions sur une variété riemannienne complète. CR Acad. Sci. Paris Sér. I. Math. 303(1): 23–26
Bakry, D.: Transformations de Riesz pour les semigroupes symétriques. II. Étude sous la condition Γ2 ≥ 0. Séminaire de Probab., XIX, pp. 145–174. Lecture Notes in Mathematics, 1123, Springer, Berlin (1985)
Bakry, D.: Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée. Séminaire de Probab., XXI, pp. 137–172. Lecture Notes in Mathematics, 1247, Springer, Berlin (1987)
Bakry, D.: La propriété de sous-harmonicité des diffusions dans les variétés. Séminaire de Probab., XII, pp. 1–50. Lecture Notes in Mathematics, 1372, Springer, Berlin (1989)
Bakry, D.: The Riesz transforms associated with second order differential operators. Seminar on Stochastic Processes (Gainesville, FL, 1988), pp. 1–43. Progr. Probab., 17, Birkhauser, Boston, MA (1989)
Bakry, D., Emery, M.: Diffusion hypercontractives. Séminaire de Probab., XIX, pp. 179–206. Lect. Notes in Math., 1123, Springer, Berlin (1985)
Bass R.F. (1995). Probabilistic Techniques in Analysis. Springer, Heidelberg
Burkholder D.L. (1984). Boundary value problems and sharp inequalities for martingale transforms. Ann. Probab. 12: 647–702
Burkholder D.L. (1987). A sharp and strict L p-inequality for stochastic integrals. Ann. Probab. 15: 268–273
Burkholder, D.L.: Martingales and Fourier analysis in Banach spaces. Probability and Analysis (Varenna, 1985), 1206, pp. 61–108, Lecture Notes in Math., Springer, Berlin (1986)
Bañuelos R. and Lindeman A. (1997). A martingale study of the Beurling–Ahlfors transforms in \({\mathbb{R}}^n\) J. Funct. Anal. 145: 224–265
Bañuelos R. and Wang G. (1995). Sharp inequalities for martingales with applications to the Beurling-Ahlfors and Riesz transforms. Duke Math. J. 80: 575–600
Calderón A.P. and Zygmund Z. (1952). On the existence of certain singular integrals. Acta Math. 88: 85–139
Chen, J.-C.: PhD Thesis, Hangzhou University, 1987
Chen, J.-C., Luo, C.: Duality of H 1 and BMO on positively curved manifolds and their characterizations. Harmonic Analysis (Tianjin, 1988), pp. 23–38. Lect. Notes in Math., 1494, Springer, Berlin (1991)
Coulhon T. and Duong X.T. (1999). Riesz transforms for 1 ≤ p ≤ 2. Trans. Am. Math. Soc. 351(3): 1151–1169
Coulhon T. and Duong X.T. (2003). Riesz transform and related inequalities on noncompact Riemannian manifolds. Commun. Pure Appl. Math. 56(12): 1728–1751
Dellacherie, C., Maisonneuve, B., Meyer, P.A.: Probabilies et Potentiel, Chap. XVII a XXIV. Processus de Markov (fin) Complements de Calcul Stochastiques, Hermann, 1992
Donati-Martin C. and Yor M. (1997). Some Brownian functionals and their laws. Ann. Probab. 25: 1011–1058
Donaldson S. and Sullivan D. (1989). Quasiconformal 4-manifolds. Acta Math. 163: 181–252
Elworthy, D., LeJan, Y., Li, X.-M.: On the Geometry of Diffusion Operators and Stochastic Flows. Lect. Notes in Math., 1720, Springer, Berlin (1999)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order, 3rd edn. Springer, Heidelberg (2001)
Gundy, R.F.: Sur les transformations de Riesz pour le semi-groupe d’Ornstein–Uhlenbeck. CR Acad. Sci. Paris Sér. I. Math. 303(19), 967–970 (1986). MR08771
Gundy, R.F.: Some Topics in Probability and Analysis. Am. Math. Soc., 70. Providence, Rhode Island (1986)
Gundy R.F. and Varopoulos N.Th. (1979). Les transformations de Riesz et les intégrales stochastiques. CR Acad. Sci. Paris Sér. I. Math. 289: A13–A16
Gundy, R.F., Silverstein, M.L.: On a probabilistic interpretation for the Riesz transforms. Functional Analysis in Markov Processes (Katata/Kyoto, 1981), pp. 199–203, Lect. Notes in Math., 923, Springer, Berlin (1982)
Iwaniec T. and Martin G. (1993). Quasiregular mappings in even dimensions. Acta Math. 170: 29–81
Iwaniec T. and Martin G. (1996). Riesz transforms and related singular integrals. J. Reine Angew. Math. 473: 25–57
Larsson-Cohn L. (2002). On the constants in the Meyer inequality. Monatsh. Math. 137: 51–56
Li J.-Y. (1991). Gradient estimate for the heat kernel of a complete Riemannian manifold and its applications. J. Funct. Anal. 97: 293–310
Li X.-D. (2005). Liouville theorems for symmetric diffusion operators on complete Riemannian manifolds. J. Math. Pure Appl. 84(10): 1295–1361
Li X.-D. (2006). Riesz transforms for symmetric diffusion operators on complete Riemannian manifolds. Rev. Math. Iberoam. 22(2): 591–648
Li, X.-D.: Riesz transforms on forms and L p-Hodge decomposition theory on complete Riemannian manifolds, preprint (2006)
Lions, P.-L.: Mathematical Topics in Fluid Mechanics. Vol. 1: Incompressible Methods, and vol. 2: Compressible Models. Oxford Univ. Press, New York (1996, 1998)
Lions P.-L. and Masmoudi N. (2001). Uniqueness of mild solutions of the Navier–Stokes system in L N. Commun. Part. Different. Equ. 26: 2211–2226
Lohoué N. (1985). Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive. J. Funct. Anal. 61(2): 164–201
Lott J. (2003). Some geometric properties of the Bakry–Emery Ricci tensor. Comment Math. Helv. 78: 865–883
Malliavin P. (1974). Formule de la moyenne, calcul des perturbations et théorèmes d’annulation pour les formes harmoniques. J. Funct. Anal. 17: 274–291
Malliavin, P.: Stochastic Analysis. Springer, Berlin (1997). MR1450093
Meyer, P.A.: Démonstrations probabilistes des inégalités de Littlewood–Paley. Séminaire de Probab., XV., pp. 179–193. Lect. Notes in Math., 1059, Springer, Berlin (1976)
Meyer, P.A.: Le dual de \(H^1({\mathbb{R}}^{\nu})\) : Démonstration probabiliste. Séminaire de Probab., XI, pp. 132–195. Lect. Notes in Math., 581, Springer, Berlin (1977)
Meyer, P.A.: Transformations de Riesz pour les lois gaussiennes. Séminaire de Probab., XVIII, pp. 179–193. Lect. Notes in Math., 1059, Springer, Berlin (1984)
Norris, J.R.: A complete differential formalism for stochastic calculus in manifolds. Séminaire de Probab., XXVI, pp. 189–209, Lecture Notes in Math., 1526, Springer, Berlin (1992)
Pichorides S.K. (1972). On the best values of the constants in the theorems of M. Riesz, Zygmund and Kolmogorov.. Stud. Math. 44: 165–179
Pisier, G.: Riesz transforms: a simple analytic proof of P.A. Meyer’s inequality. Séminaire de Probab., XXII, pp. 481–501. Lect. Notes in Math., 1321, Springer, Berlin (1988)
Rogers, L.C.G., Williams, D.: Diffusions, Markov Processes and Martingales, vol. 1. Foundations, 2nd edn. Cambridge Math. Library, New York (2000)
Revuz D. and Yor M. (2000). Continuous Martingales and Brownian Motion. Springer, Heidelberg
Riesz M. (1927). Sur les fonctions conjuguées. Math. Zeit. 27: 218–244
Shigekawa I. and Yoshida N. (1992). Littlewood–Paley–Stein inequality for a symmetric diffusion. J. Math. Soc. Jpn. 44: 251–280
Song, S.: C-semigroups on Banach spaces and functional inequalities. Séminaire de Probab., XXIX, pp. 297–326. Lect. Notes in Math., 1613, Springer, Berlin (1995)
Strichartz R. (1983). Analysis of the Laplacian on the complete Riemannian manifold. J. Funct. Anal. 52: 48–79
Stein E.M. (1970). Topics in Harmonic Analysis Related to the Littlewood–Paley Theory. Princeton University Press, Princeton
Stein E.M. (1970). Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton
Stroock D.W. and Varadhan S.R.S. (1979). Multidimensional Diffusion Processes. Springer, Heidelberg
Yoshida N. (1994). The Littlewood–Paley–Stein inequality on an infinite dimensional manifold. J. Funct. Anal. 122(2): 402–427
Yoshida N. (1992). Sobolev spaces on a Riemannian manifold and their equivalence. J. Math. Kyoto Univ. 32: 621–654
Author information
Authors and Affiliations
Corresponding author
Additional information
Dedicated to my daughter Yun-Xuan.
Research partially supported by a Delegation in CNRS at the University of Paris-Sud during the 2005–2006 academic year.
An erratum to this article is available at http://dx.doi.org/10.1007/s00440-014-0560-1.
Rights and permissions
About this article
Cite this article
Li, XD. Martingale transforms and L p-norm estimates of Riesz transforms on complete Riemannian manifolds. Probab. Theory Relat. Fields 141, 247–281 (2008). https://doi.org/10.1007/s00440-007-0085-y
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00440-007-0085-y