$L^p$ Boundedness of the Riesz transform related to Schrödinger operators on a manifold

Badr, Nadine; Ben Ali, Besma

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 8 (2009) no. 4, pp. 725-765.

We establish various $L^p$ estimates for the Schrödinger operator $-\Delta +V$ on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where $\Delta$ is the Laplace-Beltrami operator and $V$ belongs to a reverse Hölder class. At the end of this paper we apply our result to Lie groups with polynomial growth.

Classification : 35J10, 42B37

[1] 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. | MR | Zbl

[2] P. Auscher, On $L^p$ estimates for square roots of second order elliptic operators on ${ℝ}^n$, Publ. Mat. 48 (2004), 159–186. | EuDML | MR | Zbl

[3] P. Auscher and B. Ben Ali, Maximal inequalities and Riesz transform estimates on $L^p$ spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier (Grenoble) 57 (2007), 1975–2013. | EuDML | Numdam | MR | Zbl

[4] P. Auscher and T. Coulhon, Riesz transform on manifolds and Poincaré inequalities, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (5) 4 (2005), 531–555. | EuDML | Numdam | MR | Zbl

[5] P. Auscher, X.T. Duong and A. Mcintosh, Boundedness of Banach space valued singular integral operators and applications to Hardy spaces, unpublished manuscript.

[6] P. Auscher and J. M. Martell, Weighted norm inequalities, off diagonal estimates and elliptic operators. Part III: Harmonic analysis of elliptic operators, J. Funct. Anal. 241 (2006), 703–746. | MR | Zbl

[7] P. Auscher and J. M. Martell, Weighted norm inequalities, off diagonal estimates and elliptic operators. Part I: General operator theory and weights, Adv. Math. 212 (2007), 225–276. | MR | Zbl

[8] N. Badr, Real interpolation of Sobolev spaces associated to a weight, Potential Anal. 3 (2009), 345–374. | MR | Zbl

[9] D. Bakry, Etude des transformations de Riesz dans les variétés Riemanniennes à courbure de Ricci minorée, In: “Séminaire de Probabilités”, XXI, Lecture Notes in Math., Vol. 1247, Springer, Berlin, 1987, 137–172. | EuDML | Numdam | MR | Zbl

[10] S. Bockner, Vector fields and Ricci curvature, Bull. Amer. Mat. Soc. 52 (1946), 776–797. | MR | Zbl

[11] M. Braverman, O. Milatovic and M. Shubin, Essential self-adjointness of Schrödinger-type operators on manifolds, Russ. Math. Surveys 57 (2002), 641–692. | MR | Zbl

[12] S. Buckley, P. Koskela and G. Lu, Subelliptic Poincaré inequalities: The case $p<1$, Publ. Mat. 39 (1995), 313–334. | EuDML | MR | Zbl

[13] P. Buser, A note on the isoperimetric constant, Ann. Sci. École Norm. Sup. (4) (1982), 213–230. | EuDML | Numdam | MR | Zbl

[14] A. Carbonaro, G. Metafune and C. Spina, Parabolic Schrödinger operators, J. Math. Anal. Appl. 343 (2008), 965–974. | MR | Zbl

[15] I. Chavel, “Riemannian Geometry – A Modern Introduction”, Cambridge University Press, 1993. | MR | Zbl

[16] J. Cheeger, M. Gromov and M. Taylor, Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemmanian manifolds, J. Differential Geom. 17 (1982), 15–33. | MR

[17] J. C. Chen A note on Riesz potentials and the first eigenvalue, Proc. Amer. Math. Soc. 117 (1993), 683–685. | MR | Zbl

[18] R. Coifman and G. Weiss, “Analyse Harmonique sur Certains Espaces Homogènes”, Lecture Notes in Math., Springer, 1971. | Zbl

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

[20] T. Coulhon and X. T. Duong, Riesz transforms for $1<p<2$, Trans. Amer. Math. Soc. 351 (1999), 1151–1169. | MR | Zbl

[21] T. Coulhon and H. Q. Li, Estimations inférieures du noyau de la chaleur sur les variétés coniques et transformée de Riesz, Arch. Math. 83 (2004), 229–242. | MR

[22] T. Coulhon and L. Saloff-Coste, Isopérimétrie pour les groupes et les variétés, Rev. Mat. Iberoamericana 9 (1993), 293–314. | EuDML | MR

[23] T. Coulhon, L. Saloff-Coste and N. Varopoulos, “Analysis and Geometry on Groups”, Cambridge Tracts in Mathematics, 1993. | MR | Zbl

[24] M. G. Cowling, Harmonic analysis on semigroups, Ann. of Math. (2) 117 (1983), 267–283. | MR | Zbl

[25] X. T. Duong and A. Mcintosh, Singular integral operators with non-smooth kernels on irregular domains, Rev. Mat. Iberoamericana 15 (1999), 233–265. | EuDML | MR | Zbl

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

[27] T. Gallouët and J.-M. Morel, Resolution of a semilinear equation in $L^1$, Proc. Roy. Soc. Edinburgh Sect. A 96 (1984), 275–288. | MR | Zbl

[28] J. García-Cuerva and J. L. Rubio De Francia, “Weighted Norm Inequalities and Related Topics”, North Holland Math. Studies 116, North Holland, Amsterdam, 1985. | MR | Zbl

[29] A. Grigory’An, The heat equation on non compact Riemannian manifolds, Math. USSR. Sb 72 (1992), 47–76. | MR

[30] A. Grigory’An and L. Saloff Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) 3 (2005), 825–890. | EuDML | Numdam | MR | Zbl

[31] D. Guibourg, Inégalités maximales pour l’opérateur de Schrödinger C. R. Acad. Sci. Paris Sér. I Math. 316 (1993), 249–252. | MR | Zbl

[32] C. Guillarmou and A. Hassell, Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I, Math. Ann. 341 (2008), 859–896. | MR | Zbl

[33] Y. Guivarc’H, Croissance polynomiale et période des fonctions harmoniques, Bull. Soc. Math. France 101 (1973), 333–379. | EuDML | Numdam | MR | Zbl

[34] P. Hajlasz and P. Koskela, Sobolev met Poincaré, Mem. Amer. Math. Soc. 145 (2000), 1–101. | MR | Zbl

[35] B. Helffer and J. Nourrigat, Une inégalité $L^2$, unpublished manuscript.

[36] R. Johnson and C. J. Neugebauer, Change of variable results for $A_p$ and reverse Hölder ${RH}_r$ classes, Trans. Amer. Math. Soc. 328 (1991), 639–666. | Zbl

[37] T. Kato, $L^p$-theory of Schrödinger operators with a singular potential, In: “Aspects of Positivity in Function Analysis”, North-Holland Math. Stud., 1985, 63–78. | MR

[38] S. Keith and K. Rajala, A remark on Poincaré inequality on metric spaces, Math. Scand. 95 (2004), 299–304. | MR | Zbl

[39] C. Le Merdy, On square functions associated to sectorial operators, Bull. Soc. Math. France 132 (2004), 137–156. | EuDML | Numdam | MR | Zbl

[40] H. Q. Li, Ph.D thesis, 1998.

[41] H. Q. Li, Estimations $L^p$ des opérateurs de Schrödinger sur les groupes nilpotents, J. Funct. Anal. 161 (1999), 151–218. | MR

[42] H. Q. Li, La transformée de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999), 145–238. | MR

[43] H. Q. Li, Estimations du noyau de la chaleur sur les variétés coniques et ses applications, Bull. Sci. Math. 124 (2000), 365–384. | MR

[44] P. Li, “Lecture Notes on Geometric Analysis”, Lecture Notes series 6, Seoul National University, Research Institute of Mathematics, Global Analysis Research Center, Seoul, 1993. | MR | Zbl

[45] P. A. Meyer, Démonstration probabiliste de certaines inégalités de Littlewood-Paley, In: “Séminaire de Probabilités”, X, Lecture Notes in Math., Vol. 511, Springer-Verlag, 1976, 125–183. | Numdam | MR | Zbl

[46] A. Nagel, E. M. Stein and S. Wainger, Balls and metrics defined by vector fields, Acta Math. 155 (1985), 103–147. | MR | Zbl

[47] E. M. Ouhabaz, The spectral bound and principal eigenvalues of Schrödinger operators on Riemannian manifolds, Duke Math. J. 110 (2001), 1–35. | MR | Zbl

[48] L. Saloff-Coste, A note on Poincaré, Sobolev and Harnack inequalities, Internat. Math. Res. Notices 2 65 (1992), 27–38. | MR | Zbl

[49] L. Saloff-Coste, Parabolic Harnack inequality for divergence form second order differential operator, Potential Anal. 4 (1995), 429–467. | MR | Zbl

[50] L. Saloff-Coste, “Aspects of Sobolev-type Inequalities”, Cambridge University Press, 2002. | MR | Zbl

[51] Z. Shen, $L^p$ estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier (Grenoble) 45 (1995), 513–546. | EuDML | Numdam | MR | Zbl

[52] A. Sikora, Riesz transform, gaussian bounds and the method of wave equation, Math. Z. 247 (2004), 643–662. | MR | Zbl

[53] A. Sikora and J. Wright, Imaginary powers of Laplace operators, Proc. Amer. Math. Soc. 129 (2001), 1745–1754. | MR | Zbl

[54] E. M. Stein and G. Weiss, “Introduction to Fourier Analysis in Euclidean Spaces”, Princeton University Press, 1971. | Zbl

[55] E. M. Stein, “Singular Integrals and Differentiability Properties of Functions”, Princeton University Press, 1970. | MR | Zbl

[56] E. M. Stein, “Topics in Harmonic Analysis Related to the Littlewood-Paley Theory”, Princeton U.P, 1970. | MR | Zbl

[57] J. O. Strömberg and A. Torchinsky, “Weighted Hardy Spaces”, Lecture Notes in Math., Vol. 1381, Springer-Verlag, 1989. | MR | Zbl

[58] N. Varopoulos, Fonctions harmoniques sur les groupes de Lie, C. R. Acad. Sci. Paris, Sér. I Math., 304 (1987), 519–521. | MR | Zbl

[59] N. Yosida, Sobolev spaces on a Riemannian manifold and their equivalence, J. Math. Kyoto Univ. 32 (1992), 621–654. | MR | Zbl