Heat kernel and Riesz transform of Schrödinger operators
Annales de l'Institut Fourier, Volume 69 (2019) no. 2, pp. 457-513.

The goal of this article is two-fold: in the first part, we give a purely analytic proof of the Gaussian estimates for the heat kernel of Schrödinger operators Δ+𝒱 whose potential 𝒱 is “small at infinity” in a (weak) integral sense. Our results improve known results that have been proved by probabilistic techniques, and shed light on the hypotheses that are required on the potential 𝒱. In a second part, we prove sharp boundedness results for the Riesz transform with potential d(Δ+𝒱) -1/2 . A very simple characterization of p-non-parabolicity in terms of lower bounds for the volume growth, which is of independent interest, is also presented.

Le but de cet article est double : dans une première partie, nous donnons une preuve analytique des estimées Gaussiennes pour un opérateur de Schrödinger Δ+𝒱 dont le potential 𝒱 est « petit à l’infini » en un sens (faible) intégral. Nos résultats améliorent des résultats connus, qui avaient été prouvés précédemment par des techniques probabilistes, et éclairent les hypothèses qui doivent être faites sur le potentiel 𝒱. Dans une seconde partie, nous prouvons des résultats optimaux concernant l’action de la transformée de Riesz avec potentiel d(Δ+𝒱) -1/2 sur les espaces L p . Une charactérisation particulièrement simple de la non-p-parabolicité en terme de borne inférieure de la croissance du volume, qui a un intérêt en tant que tel, est aussi obtenue.

Received:
Revised:
Accepted:
Published online:
DOI: 10.5802/aif.3249
Classification: 35Kxx, 31Exx, 58Jxx
Keywords: Heat kernel, Schrödinger operators, Riesz transform, $p$-non-parabolicity.
Mot clés : Noyau de la chaleur, opérateurs de Schrödinger, transformée de Riesz, $p$-non-parabolicité

Devyver, Baptiste 1

1 Technion, Israel Institute of Technology Dept. of mathematics Haifa (Israel)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
@article{AIF_2019__69_2_457_0,
     author = {Devyver, Baptiste},
     title = {Heat kernel and {Riesz} transform of {Schr\"odinger} operators},
     journal = {Annales de l'Institut Fourier},
     pages = {457--513},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {69},
     number = {2},
     year = {2019},
     doi = {10.5802/aif.3249},
     zbl = {07067410},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3249/}
}
TY  - JOUR
AU  - Devyver, Baptiste
TI  - Heat kernel and Riesz transform of Schrödinger operators
JO  - Annales de l'Institut Fourier
PY  - 2019
SP  - 457
EP  - 513
VL  - 69
IS  - 2
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3249/
DO  - 10.5802/aif.3249
LA  - en
ID  - AIF_2019__69_2_457_0
ER  - 
%0 Journal Article
%A Devyver, Baptiste
%T Heat kernel and Riesz transform of Schrödinger operators
%J Annales de l'Institut Fourier
%D 2019
%P 457-513
%V 69
%N 2
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3249/
%R 10.5802/aif.3249
%G en
%F AIF_2019__69_2_457_0
Devyver, Baptiste. Heat kernel and Riesz transform of Schrödinger operators. Annales de l'Institut Fourier, Volume 69 (2019) no. 2, pp. 457-513. doi : 10.5802/aif.3249. https://aif.centre-mersenne.org/articles/10.5802/aif.3249/

[1] Ancona, Alano First eigenvalues and comparison of Green’s functions for elliptic operators on manifolds or domains, J. Anal. Math., Volume 72 (1997), pp. 45-92 | DOI | MR | Zbl

[2] Assaad, Joyce Riesz transforms associated to Schrödinger operators with negative potentials, Publ. Mat., Barc., Volume 55 (2011) no. 1, pp. 123-150 | DOI | MR | Zbl

[3] Assaad, Joyce; Ouhabaz, El Maati Riesz transforms of Schrödinger operators on manifolds, J. Geom. Anal., Volume 22 (2012) no. 4, pp. 1108-1136 | DOI | Zbl

[4] Auscher, Pascal; Ben-Ali, Besma Maximal inequalities and Riesz transform estimates on L p spaces for Schrödinger operators with nonnegative potentials, Ann. Inst. Fourier, Volume 57 (2007) no. 6, pp. 1975-2013 | DOI | Zbl

[5] Auscher, Pascal; Coulhon, Thierry; Duong, Xuan Thinh; Hofmann, Steve Riesz transform on manifolds and heat kernel regularity, Ann. Sci. Éc. Norm. Supér., Volume 37 (2004) no. 6, pp. 911-957 | DOI | MR | Zbl

[6] Badr, Nadine; Ben-Ali, Besma L p boundedness of the Riesz transform related to Schrödinger operators on a manifold, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 8 (2009) no. 4, pp. 725-765 | Zbl

[7] Bakry, Dominique Étude des Transformations de Riesz dans les variétés Riemanniennes à courbure de Ricci minorée, Séminaire de probabilités XIX (1983/84) (Lecture Notes in Mathematics), Volume 1123, Springer, 1985, pp. 145-174 | Zbl

[8] Boutayeb, Salahaddine; Coulhon, Thierry; Sikora, Adam A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces, Adv. Math., Volume 270 (2015), pp. 302-374 | DOI | MR | Zbl

[9] Buser, Peter A note on the isoperimetric constant, Ann. Sci. Éc. Norm. Supér., Volume 15 (1982) no. 2, pp. 213-230 | DOI | MR | Zbl

[10] Carron, Gilles Geometric inequalities for manifolds with Ricci curvature in the Kato class (2016) (https://arxiv.org/abs/1612.03027) | Zbl

[11] Carron, Gilles Riesz transform on manifolds with quadratic curvature decay (2016) (https://arxiv.org/abs/1403.6278) | Zbl

[12] Carron, Gilles; Coulhon, Thierry; Hassell, Andrew Riesz transform and L p -cohomology for manifolds with Euclidean ends, Duke Math. J., Volume 133 (2006) no. 1, pp. 59-93 | DOI | MR | Zbl

[13] Cheeger, Jeff; Gromov, Mikhail; Taylor, Michael Finite propagation speed, kernel estimates for functions of the Laplace operator, and the geometry of complete Riemannian manifolds, J. Differ. Geom., Volume 17 (1982) no. 1, pp. 15-53 | MR | Zbl

[14] Chen, Peng; Magniez, Jocelyn; Ouhabaz, El Maati Riesz transforms on non-compact manifolds (2014) (https://arxiv.org/abs/1411.0137) | Zbl

[15] Chen, Zhen-Qing Gaugeability and conditional gaugeability, Trans. Am. Math. Soc., Volume 354 (2002) no. 11, pp. 4639-4679 | DOI | MR | Zbl

[16] Coulhon, Thierry Dimensions at infinity for Riemannian manifolds, Potential Anal., Volume 4 (1995) no. 4, pp. 335-344 | DOI | MR | Zbl

[17] Coulhon, Thierry; Devyver, Baptiste; Sikora, Adam Gaussian heat kernel estimates: from functions to forms (2016) (https://arxiv.org/abs/1606.02423)

[18] Coulhon, Thierry; Dungey, Nicholas Riesz transform and perturbation, J. Geom. Anal., Volume 17 (2007) no. 2, pp. 213-226 | DOI | MR | Zbl

[19] Coulhon, Thierry; Duong, Xuan Thinh Riesz transforms for 1p2, Trans. Am. Math. Soc., Volume 351 (1999) no. 3, pp. 1151-1169 | DOI | Zbl

[20] Coulhon, Thierry; Duong, Xuan Thinh Riesz transform and related inequalities on noncompact Riemannian manifolds, Commun. Pure Appl. Math., Volume 56 (2003) no. 12, pp. 1728-1751 | DOI | MR | Zbl

[21] Coulhon, Thierry; Holopainen, Ilkka; Saloff-Coste, Laurent Harnack inequality and hyperbolicity for subelliptic p-Laplacians with applications to Picard type theorems, Geom. Funct. Anal., Volume 11 (2001) no. 6, pp. 1139-1191 erratum in ibid. 12 (2002), no. 11, p. 217 | DOI | MR | Zbl

[22] Coulhon, Thierry; Saloff-Coste, Laurent Variétés Riemanniennes isométriques à l’infini, Rev. Mat. Iberoam., Volume 11 (1995) no. 3, pp. 687-726 | DOI | Zbl

[23] Coulhon, Thierry; Saloff-Coste, Laurent; Varopoulos, Nicholas Analysis and geometry on groups, Cambridge Tracts in Mathematics, 100, Cambridge University Press, 1992 | MR

[24] Coulhon, Thierry; Zhang, Qi S. Large time behavior of heat kernels on forms, J. Differ. Geom., Volume 77 (2007) no. 3, pp. 353-384 | DOI | MR | Zbl

[25] Davies, Edward Brian; Simon, Barry L p norms of noncritical Schrödinger semigroups, J. Funct. Anal., Volume 102 (1991) no. 1, pp. 95-115 | Zbl

[26] Devyver, Baptiste A Gaussian estimate for the heat kernel on differential forms and application to the Riesz transform, Math. Ann., Volume 358 (2014) no. 1-2, pp. 25-68 | DOI | MR | Zbl

[27] Devyver, Baptiste A perturbation result for the Riesz transform, Ann. Sc. Norm. Super. Pisa, Cl. Sci., Volume 14 (2015) no. 3, pp. 937-964 | MR | Zbl

[28] Devyver, Baptiste; Fraas, Martin; Pinchover, Yehuda Optimal Hardy weight for second-order elliptic operator: an answer to a problem of Agmon, J. Funct. Anal., Volume 266 (2014) no. 7, pp. 4422-4489 | DOI | MR | Zbl

[29] Ganguly, Debdip; Pinchover, Yehuda On the equivalence of heat kernels of second-order parabolic operators (2016) (https://arxiv.org/abs/1606.08601) | Zbl

[30] Grigoryan, Alexander Heat kernel upper bounds on a complete non-compact manifold, Rev. Mat. Iberoam., Volume 10 (1994) no. 2, pp. 395-452 | DOI | MR | Zbl

[31] Grigoryan, Alexander Analytic and geometric background of recurrence and non-explosion of the Brownian motion on Riemannian manifolds, Bull. Am. Math. Soc., Volume 36 (1999) no. 2, pp. 135-249 | DOI | MR | Zbl

[32] Grigoryan, Alexander Heat kernels on weighted manifolds and applications, The ubiquitous heat kernel (Contemporary Mathematics), Volume 398, American Mathematical Society, 2006, pp. 93-191 | DOI | MR | Zbl

[33] Grigoryan, Alexander Heat Kernel and Analysis on Manifolds, AMS/IP Studies in Advanced Mathematics, 47, American Mathematical Society; International Press, 2009, xvii+482 pages | MR | Zbl

[34] Grigoryan, Alexander; Hu, Jiaxin; Lau, Ka-Sing Heat kernels on metric spaces with doubling measure, Fractal geometry and stochastics IV (Progress in Probability), Volume 61, Birkhäuser, 2009, pp. 3-44 | DOI | MR | Zbl

[35] Guillarmou, Colin; Hassell, Andrew Resolvent at low energy and Riesz transform for Schrödinger operators on asymptotically conic manifolds. I, Math. Ann., Volume 341 (2008) no. 4, pp. 859-896 | Zbl

[36] Holopainen, Ilkka Volume growth, Green’s functions, and parabolicity of ends, Duke Math. J., Volume 97 (1999) no. 2, pp. 319-346 | DOI | MR | Zbl

[37] LeBrun, Claude Complete Ricci-flat Kähler metrics on n need not be flat, Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989) (Proceedings of Symposia in Pure Mathematics), Volume 52, American Mathematical Society, 1991, pp. 297-304 | DOI | Zbl

[38] Li, Peter Geometric analysis, Cambridge Studies in Advanced Mathematics, 134, Cambridge University Press, 2012, x+406 pages | MR | Zbl

[39] Murata, Minoru Semismall perturbations in the Martin theory for elliptic equations, Isr. J. Math., Volume 102 (1997), pp. 29-60 | DOI | MR | Zbl

[40] Pinchover, Yehuda On positive solutions of second-order elliptic equations, stability results, and classification, Duke Math. J., Volume 57 (1988) no. 3, pp. 955-980 | MR | Zbl

[41] Pinchover, Yehuda Criticality and ground states for second-order elliptic equations, J. Differ. Equations, Volume 80 (1989) no. 2, pp. 237-250 | DOI | MR | Zbl

[42] Pinchover, Yehuda On criticality and ground states of second order elliptic equations. II, J. Differ. Equations, Volume 87 (1990) no. 2, pp. 353-364 | MR | Zbl

[43] Pinchover, Yehuda On the equivalence of Green functions of second order elliptic equations in n , Differ. Integral Equ., Volume 5 (1992) no. 3, pp. 481-493 | MR | Zbl

[44] Pinchover, Yehuda Maximum and anti-maximum principles and eigenfunctions estimates via perturbation theory of positive solutions of elliptic equations, Math. Ann., Volume 314 (1999) no. 3, pp. 555-590 | MR | Zbl

[45] Pinchover, Yehuda Topics in the theory of positive solutions of second-order elliptic and parabolic partial differential equations, Spectral theory and mathematical physics: a Festschrift in honor of Barry Simon’s 60th birthday (Proceedings of Symposia in Pure Mathematics), Volume 76, American Mathematical Society, 2007, pp. 329-355 | MR | Zbl

[46] Pinchover, Yehuda; Tintarev, Kyril A ground state alternative for singular Schrödinger operators, J. Funct. Anal., Volume 230 (2006) no. 1, pp. 65-77 | Zbl

[47] Pinchover, Yehuda; Tintarev, Kyril Ground state alternative for p-Laplacian with potential term, Calc. Var. Partial Differ. Equ., Volume 28 (2007) no. 2, pp. 179-201 | MR | Zbl

[48] Saloff-Coste, Laurent Aspects of Sobolev-type inequalities, London Mathematical Society Lecture Note Series, 289, Cambridge University Press, 2002, x+190 pages | MR | Zbl

[49] Shen, Zhong Wei L p estimates for Schrödinger operators with certain potentials, Ann. Inst. Fourier, Volume 45 (1995) no. 2, pp. 513-546 | DOI | MR | Zbl

[50] Sikora, Adam Riesz transform, Gaussian bounds and the method of wave equation, Math. Z., Volume 247 (2004) no. 3, pp. 643-662 | MR | Zbl

[51] Simon, Barry Brownian motion, L p properties of Schrödinger operators and the localization of binding, J. Funct. Anal., Volume 35 (1980) no. 2, pp. 215-229 | MR | Zbl

[52] Takeda, Masayoshi Conditional gaugeability and subcriticality of generalized Schrödinger operators, J. Funct. Anal., Volume 191 (2002) no. 2, pp. 343-376 | MR | Zbl

[53] Takeda, Masayoshi Gaussian bounds of heat kernels for Schrödinger operators on Riemannian manifolds, Bull. Lond. Math. Soc., Volume 39 (2007) no. 1, pp. 85-94 | DOI | MR | Zbl

[54] Zhang, Qi S. On a parabolic equation with a singular lower order term. II. The Gaussian bounds, Indiana Univ. Math. J., Volume 46 (1997) no. 3, pp. 989-1020 | MR | Zbl

[55] Zhang, Qi S. Global bounds of Schrödinger heat kernels with negative potentials, J. Funct. Anal., Volume 182 (2001) no. 2, pp. 344-370 | Zbl

[56] Zhang, Qi S. A sharp comparison result concerning Schrödinger heat kernels, Bull. Lond. Math. Soc., Volume 35 (2003) no. 4, pp. 461-472 | DOI | MR | Zbl

[57] Zhao, Zhongxin Subcriticality and gaugeability of the Schrödinger operator, Trans. Am. Math. Soc., Volume 334 (1992) no. 1, pp. 75-96 | Zbl

Cited by Sources: