Heat kernel and Riesz transform of Schrödinger operators

Devyver, Baptiste

Annales de l'Institut Fourier, Tome 69 (2019) no. 2, pp. 457-513.

Le but de cet article est double : dans une première partie, nous donnons une preuve analytique des estimées Gaussiennes pour un opérateur de Schrödinger $\Delta +𝒱$ 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 $\mathrm{d}{(\Delta +𝒱)}^{-1/2}$ 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.

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 $\Delta +𝒱$ 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 $\mathrm{d}{(\Delta +𝒱)}^{-1/2}$. 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.

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

[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

[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)

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

[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)

[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 $1\le p\le 2$, 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)

[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 ${ℂ}^n$ 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 ${ℝ}^n$, 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 | 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 | Zbl

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