Riesz transform on manifolds and Poincaré inequalitie

Auscher, Pascal; Coulhon, Thierry

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 4 (2005) no. 3, pp. 531-555.

We study the validity of the $L^p$ inequality for the Riesz transform when $p>2$ and of its reverse inequality when $1<p<2$ on complete riemannian manifolds under the doubling property and some Poincaré inequalities.

Classification : 58J35, 42B20

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

[2] P. Auscher, T. Coulhon, X. T. Duong and S. Hofmann, Riesz transforms on manifolds and heat kernel regularity, Ann. Sci. École Norm. Sup. 37 (2004), 911-95. | Numdam | MR | Zbl

[3] P. Auscher and J.-M. Martell, Weighted norm inequalities off-diagonal estimates and elliptic operators: Part I, preprint 2005. | MR

[4] P. Auscher and P. Tchamitchian, “Square Root Problem for Divergence Operators and Related Topics”, Astérisque, 249, 1998. | Numdam | MR | Zbl

[5] D. Bakry, Transformations de Riesz pour les semi-groupes symétriques | Numdam | MR | Zbl

[6] S. Blunck and P. Kunstmann, Calderón-Zygmund theory for non-integral operators and the $H^{\infty }$ functional calculus, Rev. Mat. Iberoamericana 19 (2003), 919-942. | MR | Zbl

[7] G. Carron, T. Coulhon and A. Hassell, Riesz transform and $L^p$ cohomology for manifolds with Euclidean ends, Duke Math. J., to appear. | MR | Zbl

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

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

[10] T. Coulhon and X. T. Duong, Riesz transform and related inequalities on non-compact Riemannian manifolds, Comm. Pure Appl. Math. 56, 12 (2003), 1728-1751. | MR | Zbl

[11] 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 | Zbl

[12] B. Franchi, C. Pérez and R. Wheeden, Self-improving properties of John-Nirenberg and Poincaré inequalities on spaces of homogeneous type, J. Funct. Anal. 153 (1998), 108-146. | MR | Zbl

[13] F. W. Gehring, The $L^p$-integrability of the partial derivatives of a quasiconformal mapping, Acta Math. 130 (1973), 265-277. | MR | Zbl

[14] M. Giaquinta, “Multiple Integrals in the Calculus of Variations and Non-linear Elliptic Systems”, Annals of Math. Studies, vol. 105, Princeton Univ. Press, 1983. | MR | Zbl

[15] A. Grigor'Yan, Stochastically complete manifolds, (Russian) Dokl. Akad. Nauk SSSR 290 (1986), 534-537. | MR | Zbl

[16] A. Grigor'Yan, Upper bounds of derivatives of the heat kernel on an arbitrary complete manifold, J. Funct. Anal. 127 (1995), 363-389. | MR | Zbl

[17] A. Grigor'Yan and L. Saloff-Coste, Stability results for Harnack inequalities, Ann. Inst. Fourier (Grenoble) 55 (2005), 825-890. | Numdam | MR | Zbl

[18] P. Hajłasz and P. Koskela, “Sobolev Met Poincaré”, Mem. Amer. Math. Soc., Vol. 145, no. 688, 2000. | MR | Zbl

[19] S. Hofmann and J.-M. Martell, $L^p$ bounds for Riesz transforms and square roots associated to second order elliptic operators, Publ. Mat. 47 (2003), 497-515. | MR | Zbl

[20] T. Iwaniec, The Gehring lemma, In: “Quasiconformal Mappings and Analysis” (Ann. Arbor, MI, 1995), Springer, New York, 1998, 181-204. | MR | Zbl

[21] S. Keith and X. Zhong, The Poincaré inequality is an open ended condition, preprint 2003. | MR

[22] Li Hong-Quan, La transformation de Riesz sur les variétés coniques, J. Funct. Anal. 168 (1999), 145-238. | MR | Zbl

[23] J.-M. Martell, PhD thesis, Universidad Autónoma de Madrid, 2003.

[24] N. G. Meyers, An $L^p$ estimate for the gradient of solutions of second order elliptic divergence equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (3), 17 (1963), 189-206. | Numdam | MR | Zbl

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

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

[27] Z. Shen, Bounds of Riesz transforms on $L^p$ spaces for second order elliptic operators, Ann. Inst. Fourier (Grenoble) 55 (2005), 173-197. | Numdam | MR | Zbl

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

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