The Riesz transformation on conical varieties. (La transformation de Riesz sur les variétés coniques.) (French) Zbl 0937.43004

Let \(N\) be a connected Riemannian manifold of dimension \(n -1\geq 1\). This paper is concerned with the investigation of the Riesz transform \(\nabla(-\Delta)^{-1/2}\) on the cone \(C(N)= R^+\times N\). When \(N\) is compact, let \(\lambda_1\) be the smallest non-zero eigenvalue of the Laplace-Beltrami operator on \(N\) and \[ p_0 =\text{sup}\bigl\{p>1\mid p.\bigl(\tfrac n2-\sqrt{(\tfrac{n-2}{2})^2+\lambda_1}\bigr)<n\bigr\}. \] The author shows that \(\nabla(-\Delta)^{-1/2}\) is of weak type (1,1) and type \((p,p)\) for all \(1 <p <p_0\) and that it is not of type \((p,p)\) for any \(p >p_0\).
When \(N\) is a complete noncompact Riemannian manifold, the author studies the behavior of the Riesz transform on the cone \(C(N)\) in several cases. For example, if \(N =G/K\) is a Riemannian symmetric space of noncompact type then \(\nabla(-\Delta)^{-1/2}\) is bounded on \(L^p(C(N))\) for all \(1< p < n\), while if \(N= R^{n-1}\) with \(n \geq 3\), then \(\nabla(- \Delta)^{-1/2}\) is not bounded on \(L^p(C(N))\) for any \(p>n\).
Reviewer: Zhu Fulin (Hubei)


43A85 Harmonic analysis on homogeneous spaces
22E30 Analysis on real and complex Lie groups
Full Text: DOI


[1] Alexopoulos, G., An application of homogeneisation theory to harmonic analysis: Harnack inequalities and Riesz transforms on Lie groups of polynomial growth, Canad. J. Math., 44, 691-727, (1992) · Zbl 0792.22005
[2] Anker, J.-P.; Lohoué, N., Multiplicateurs sur certains espaces symétriques, Amer. J. Math., 108, 1303-1354, (1986) · Zbl 0616.43009
[3] Anker, J.-P., Sharp estimates for some functions of the Laplacian on noncompact symmetric spaces, Duke Math. J., 65, 257-297, (1992) · Zbl 0764.43005
[4] Anker, J.-P., Le noyau de la chaleur sur LES expaces symétriques U(p, q)/U(pU(q), Lecture Notes in Math., (1988), Springer-Verlag Berlin, p. 60-82
[5] Anker, J.-P.; Damek, E.; Yacoub, C., Spherical analysis on harmonic AN groups, Ann. Scuola Norm. sup. Pisa Cl. Sci., 23, 643-649, (1996) · Zbl 0881.22008
[6] J.-P. Anker, et, Lizhen, Ji, Heat kernel and Green function estimates on noncompact symmetric spaces, I, II, prépublication, 1998.
[7] Aubin, T., Espaces de Sobolev sur LES variétés riemanniennes, Bull. Sci. Math. (2), 100, 149-173, (1976) · Zbl 0328.46030
[8] Bakry, D., Transformations de Riesz pour LES semi-groupes symétriques, Séminaire de Probabilités, XIX, Lecture Notes in Math., 1123, (1985), Springer-Verlag New York, p. 130-175 · Zbl 0561.42010
[9] Bakry, D., Etude des transformations de Riesz dans LES variétés riemanniennes à courbure de Ricci minorée, Séminaire de Probabilités, XXI, Lecture Notes in Math., 1247, (1987), Springer-Verlag New York/Berlin, p. 137-172
[10] Bakry, D., The Riesz transforms associated with second order differential operators, Seminar on Stochastic Processes, 88, (1989), Birkhäuser Basel · Zbl 0689.58032
[11] Berline, N.; Getzler, E.; Vergne, M., Heat Kernels and Dirac Operators, (1992), Springer-Verlag Berlin/Heidelberg · Zbl 0744.58001
[12] Bishop, R.; Crittenden, R., Geometry of Manifolds, (1964), Academic Press New York · Zbl 0132.16003
[13] Cheeger, J.; Taylor, M. E., On the diffraction of waves by conical singularities, I, Comm. Pure Appl. Math., 25, 275-331, (1982) · Zbl 0526.58049
[14] Cheeger, J., Spectral geometry of singular Riemannian spaces, J. Differential Geom., 18, 575-657, (1983) · Zbl 0529.58034
[15] Chen, J.-C., Weak type (1, 1) boundedness of Riesz transform on positively curved manifolds, Chinese Ann. Math., 13, 1-5, (1992) · Zbl 0787.47027
[16] Cheng, S.-Y.; Li, P.; Yau, S.-T., On the upper estimate of complete Riemannian manifold, Amer. J. Math., 103, 1021-1063, (1981) · Zbl 0484.53035
[17] Coifman, R.; Weiss, G., Analyse harmonique non-commutative sur certains espaces homogènes, Lecture Notes in Math., 242, (1971), Springer-Verlag New York/Berlin
[18] Coulhon, T.; Duong, X. T., Riesz transforms for 1⩽p⩽2, Trans. Amer. Math. Soc., 351, 1151-1169, (1999) · Zbl 0973.58018
[19] Coulhon, T.; Ledoux, M., Isopérimétrie, décroissance du noyau de la chaleur et transformation de Riesz: un contre-exemple, Ark. Mat., 32, 63-77, (1994) · Zbl 0826.53035
[20] Davies, E. B.; Manpouvalos, N., Heat kernel bounds on hyperbolic space and kleinan groups, Proc. London Math. Soc., 57, 182-208, (1988) · Zbl 0643.30035
[21] Davies, E. B., Pointwise bounds on the space and time derivatives of heat kernels, J. Operator Theory, 21, 367-378, (1989) · Zbl 0702.35106
[22] Davies, E. B., Heat Kernels and Spectral Theory, (1989), Cambridge Univ. Press Cambridge · Zbl 0699.35006
[23] Gangolli, R. A.; Varadarajan, V. S., Harmonic Analysis of Spherical Functions on Real Reductive Groups, Ergeb. Math. Grenzgeb. (3), 101, (1988), Springer-Verlag Berlin · Zbl 0675.43004
[24] Grigor’yan, A., Gaussian upper bounds for the heat kernel on arbirary manifolds, J. Differential Geom., 45, 33-52, (1997) · Zbl 0865.58042
[25] Helgason, S., Differential Geometry, Lie Groups, and Symmetric Spaces, (1978), Academic Press New York · Zbl 0451.53038
[26] Li, H.-Q., La transformation de Riesz sur LES variétés coniques, C. R. Acad. Sci. Paris Sér. I Math., 326, 1167-1170, (1998) · Zbl 0910.43009
[27] H.-Q. Li, Estimations du noyau de la chaleur sur les variétés coniques et ses applications, Bull. Sci. Math, in press.
[28] Jiaya, Li, Gardient estimate for the heat kernel of a complete Riemannian manifold and its applications, J. Funct. Anal., 97, 293-310, (1991) · Zbl 0724.58064
[29] Li, P.; Yau, S.-T., On the parabolic kernel of the Schrödinger operator, Acta Math., 156, 153-201, (1986)
[30] Lohoué, N., Comparaison des champs de vecteurs et des puissances du laplacien sur une variété riemannienne à courbure non positive, J. Funct. Anal., 61, 164-201, (1985) · Zbl 0605.58051
[31] Lohoué, N., Transformées de Riesz et fonctions de littlewood – paley sur LES groupes non moyennables, C. R. Acad. Sci. Paris, 306, 327-330, (1988) · Zbl 0661.43002
[32] Lohoué, N., Estimations de certaines fonctions maximales et des transformées de Riesz multiples sur LES variétés de cartan – hadamard et LES groupes unimodulaires, C. R. Acad. Sci. Paris, 312, 561-566, (1991) · Zbl 0746.58073
[33] Lohoué, N., Transformées de Riesz et fonctions sommables, Amer. J. Math., 114, 875-922, (1992) · Zbl 0774.58040
[34] Lohoué, N.; Rychener, T., Die resolvente von δ auf symmetrischen Räumen vom nichtkompakten typ, Comment. Math. Helv., 57, 445-468, (1982) · Zbl 0505.53022
[35] Lohoué, N.; Varopoulos, N. Th., Remarques sur LES transformées de Riesz sur LES groupes de Lie nilpotents, C. R. Acad. Sci. Paris Sér. I Math., 301, 559-560, (1985) · Zbl 0582.43003
[36] Magnus, W.; Oberhettinger, F.; Soni, R. P., Formulas and Theorems for Special Functions of Mathematical Physics, (1966), Springer-Verlag Berlin/Heidelberg · Zbl 0143.08502
[37] Michel, D., Estimés des coefficients du Laplacian d’une variété riemannienne, Bull. Sci. Math. (2), 102, 15-41, (1978) · Zbl 0376.53007
[38] Saloff-Coste, L., Analyse sur LES groupes de Lie à croissance polynômiale, Ark. Math., 28, 315-331, (1990) · Zbl 0715.43009
[39] Sawyer, P., The heat equation on the spaces of positive definite matrices, Canad. J. Math., 44, 624-651, (1992) · Zbl 0772.58057
[40] Sawyer, P., On an upper bound for the heat kernel on SU*(2p)/sp(n), Canad. Math. Bull., 37, 408-418, (1994) · Zbl 0811.58056
[41] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, (1970), Princeton Univ. Press Princeton · Zbl 0207.13501
[42] Strichartz, R., Analysis of the Laplacian on the complete Riemannian manifold, J. Funct. Anal., 52, 48-79, (1983) · Zbl 0515.58037
[43] Sturm, K.-T., Heat kernel bounds on manifolds, Math. Ann., 292, 149-162, (1992) · Zbl 0747.58048
[44] Taylor, M. E., Partial Differential Equations. II. Qualitative Studies of Linear Equations, (1996), Springer-Verlag New York · Zbl 0869.35003
[45] Watson, G. N., A Treatise on the Theory of Bessel Functions, (1945), Cambridge Univ. Press Cambridge
[46] Yau, S.-T.; Schoen, R., Differentiel Geometry, (1991), Sci. Press
[47] Varopoulos, N. Th., Small time Gaussian estimates of heat diffusion kernel. I. the semigroup technique, Bull. Sci. Math., 113, 253-277, (1989) · Zbl 0703.58052
[48] Zhu, F.-L., The heat kernel of the second classical domain and of the symmetric space of a normal real form, Chinese Ann. Math. Ser. A, 13, 385-399, (1992) · Zbl 0783.43007
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