Duke Mathematical Journal

Riesz transform and $L^p$-cohomology for manifolds with Euclidean ends

Gilles Carron, Thierry Coulhon, and Rew Hassell

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Let $M$ be a smooth Riemannian manifold that is the union of a compact part and a finite number of Euclidean ends, $\mathbb{R}^n {\setminus} B(0,R)$ for some $R > 0$, each of which carries the standard metric. Our main result is that the Riesz transform on $M$ is bounded from $L^p(M) \to L^p(M; T^*M)$ for $1\lt p \lt n$ and unbounded for $p \geq n$ if there is more than one end. It follows from known results that in such a case, the Riesz transform on $M$ is bounded for $1\lt p\leq 2$ and unbounded for $p \gt n$; the result is new for $2\lt p\leq n$. We also give some heat kernel estimates on such manifolds.

We then consider the implications of boundedness of the Riesz transform in $L^p$ for some $p > 2$ for a more general class of manifolds. Assume that $M$ is an $n$-dimensional complete manifold satisfying the Nash inequality and with an $O(r^n)$ upper bound on the volume growth of geodesic balls. We show that boundedness of the Riesz transform on $L^p$ for some $p > 2$ implies a Hodge–de Rham interpretation of the $L^p$-cohomology in degree $1$ and that the map from $L^2$- to $L^p$-cohomology in this degree is injective

Article information

Duke Math. J. Volume 133, Number 1 (2006), 59-93.

First available in Project Euclid: 19 April 2006

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 58J50: Spectral problems; spectral geometry; scattering theory [See also 35Pxx]
Secondary: 58J35: Heat and other parabolic equation methods 42B20: Singular and oscillatory integrals (Calderón-Zygmund, etc.)


Carron, Gilles; Coulhon, Thierry; Hassell, Rew. Riesz transform and L p -cohomology for manifolds with Euclidean ends. Duke Math. J. 133 (2006), no. 1, 59--93. doi:10.1215/S0012-7094-06-13313-6. https://projecteuclid.org/euclid.dmj/1145452056.

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  • M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, Dover, New York, 1965.
  • G. Alexopoulos, An application of homogenization theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math. 44 (1992), 691--727.
  • J.-P. Anker, Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J. 65 (1992), 257--297.
  • P. Auscher and T. Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Sc. Norm. Super. Pisa Cl. Sci. (5) 4 (2005), 531--555.
  • P. Auscher, T. Coulhon, X. T. Duong, and S. Hofmann, Riesz transform on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. (4) 37 (2004), 911--957.
  • D. Bakry, ``Étude des transformations de Riesz dans les variétés riemanniennes à courbure de Ricci minorée'' in Séminaire de Probabilités, XXI, Lecture Notes in Math. 1247, Springer, Berlin, 1987, 137--172.
  • D. Bakry, T. Coulhon, M. Ledoux, and L. Saloff-Coste, Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033--1074.
  • I. Benjamini, I. Chavel, and E. A. Feldman, Heat kernel lower bounds on Riemannian manifolds using the old ideas of Nash, Proc. London Math. Soc. (3) 72 (1996), 215--240.
  • P. H. BéRard, From vanishing theorems to estimating theorems: The Bochner technique revisited, Bull. Amer. Math. Soc. (N.S.) 19 (1988), 371--406.
  • H.-D. Cao, Y. Shen, and S. Zhu, The structure of stable minimal hypersurfaces in $\RR^n+1$, Math. Res. Lett. 4 (1997), 637--644.
  • E. A. Carlen, S. Kusuoka, and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Probab. Statist. 23 (1987), no. 2, suppl., 245--287.
  • G. Carron, ``Inégalités isopérimétriques de Faber-Krahn et conséquences'' in Actes de la table ronde de géométrie différentielle (Luminy, France, 1992), Sémin. Congr. 1, Soc. Math. France, Montrouge, 1996, 205--232.
  • —, Cohomologie $L^2$ et parabolicité, J. Geom. Anal. 15 (2005), 391--404.
  • T. Coulhon and X. T. Duong, Riesz transforms for $1 \leq p \leq 2$, Trans. Amer. Math. Soc. 351 (1999), 1151--1169.
  • —, Riesz transform and related inequalities on noncompact Riemannian manifolds, Comm. Pure Appl. Math. 56 (2003), 1728--1751.
  • T. Coulhon and M. Ledoux, Isopérimétrie, décroissance du noyau de la chaleur et transformations de Riesz: Un contre-exemple, Ark. Mat. 32 (1994), 63--77.
  • T. Coulhon and H.-Q. Li, Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz, Arch. Math. (Basel) 83 (2004), 229--242.
  • G. De Rham, Variétés différentiables. Formes, courants, formes harmoniques, Publ. Inst. Math. Univ. Nancago 3, Actualités Sci. Indust. 1222b, 3rd ed., Hermann, Paris, 1973.
  • A. Grigor'Yan, Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoamericana 10 (1994), 395--452.
  • —, ``Estimates of heat kernels on Riemannian manifolds'' in Spectral Theory and Geometry (Edinburgh, 1998), London Math. Soc. Lecture Note Ser. 273, Cambridge Univ. Press, Cambridge, 1999, 140--225.
  • A. Grigor'Yan and L. Saloff-Coste, Heat kernel on connected sums of Riemannian manifolds, Math. Res. Lett. 6 (1999), 307--321.
  • A. Hassell and A. Vasy, Symbolic functional calculus of N-body resolvent estimates, J. Funct. Anal. 173 (2000), 257--283.
  • —, The resolvent for Laplace-type operators on asymptotically conic spaces, Ann. Inst. Fourier (Grenoble) 51 (2001), 1299--1346.
  • L. HöRmander, The Analysis of Linear Partial Differential Operators, 3: Pseudo-Differential Operators, Grundlehren Math. Wiss. 274, Springer, Berlin, 1985.
  • H.-Q. Li, La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999), 145--238.
  • N. Lohoué, Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal. 61 (1985), 164--201.
  • R. B. Melrose, Calculus of conormal distributions on manifolds with corners, Internat. Math. Res. Notices 1992, no. 3, 51--61.
  • —, The Atiyah-Patodi-Singer Index Theorem, Res. Notes in Math. 4, Peters, Wellesley, Mass., 1993.
  • —, ``Spectral and scattering theory for the Laplacian on asymptotically Euclidian spaces'' in Spectral and Scattering Theory (Sanda, Japan, 1992), Lect. Notes Pure Appl. Math. 161, Dekker, New York, 1994, 85--130.
  • E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Math. Ser. 43, Monogr. Harmon. Anal. 3, Princeton Univ. Press, Princeton, 1993.
  • N. T. Varopoulos, Hardy-Littlewood theory for semigroups, J. Funct. Anal. 63 (1985), 240--260.