Non ultracontractive heat kernel bounds by Lyapunov conditions

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March 2015, 35(3): 857-870. doi: 10.3934/dcds.2015.35.857

Non ultracontractive heat kernel bounds by Lyapunov conditions

François Bolley ^1, , Arnaud Guillin ^2, and Xinyu Wang ^3,

1.	Ceremade, Umr Cnrs 7534, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, F-75775 Paris cedex 16, France
2.	Institut Universitaire de France and Laboratoire de Mathématiques, Umr Cnrs 6620, Université Blaise Pascal, Avenue des Landais, F-63177 Aubière cedex, France
3.	School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China

Received November 2013 Revised July 2014 Published October 2014

Abstract
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Nash and Sobolev inequalities are known to be equivalent to ultracontractive properties of heat-like Markov semigroups, hence to uniform on-diagonal bounds on their kernel densities. In non ultracontractive settings, such bounds can not hold, and (necessarily weaker, non uniform) bounds on the semigroups can be derived by means of weighted Nash (or super-Poincaré) inequalities. The purpose of this note is to show how to check these weighted Nash inequalities in concrete examples of reversible diffusion Markov semigroups in $\mathbb{R}^d$, in a very simple and general manner. We also deduce off-diagonal bounds for the Markov kernels of the semigroups, refining E. B. Davies' original argument.

Keywords: Heat kernel, ultracontractivity, Nash inequality, super-Poincaré inequality., Markov semigroup.

Mathematics Subject Classification: 35P05, 47D07, 35K08, 60J6.

Citation: François Bolley, Arnaud Guillin, Xinyu Wang. Non ultracontractive heat kernel bounds by Lyapunov conditions. Discrete and Continuous Dynamical Systems, 2015, 35 (3) : 857-870. doi: 10.3934/dcds.2015.35.857

References:

[1]	D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Elec. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352.
[2]	D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, Revista Mat. Iberoam., 28 (2012), 879-906. doi: 10.4171/RMI/695.
[3]	D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grund. math. Wiss., vol. 348. Cham, 2014. doi: 10.1007/978-3-319-00227-9.
[4]	S. G. Bobkov and M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Prob., 37 (2009), 403-427. doi: 10.1214/08-AOP407.
[5]	S. Boutayeb, T. Coulhon and A. Sikora, A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces, Preprint, november 2013.
[6]	E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Prob. Stat., 23 (1987), 245-287.
[7]	P. Cattiaux, N. Gozlan, A. Guillin and C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry, Elec. J. Prob., 15 (2010), 346-385. doi: 10.1214/EJP.v15-754.
[8]	P. Cattiaux, A. Guillin, F.-Y. Wang and L. Wu, Lyapunov conditions for super Poincaré inequalities, J. Funct. Anal., 256 (2009), 1821-1841. doi: 10.1016/j.jfa.2009.01.003.
[9]	T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539. doi: 10.1006/jfan.1996.0140.
[10]	T. Coulhon, Heat kernel estimates, Sobolev-type inequalities and Riesz transform on noncompact Riemannian manifolds, In Analysis and geometry of metric measure spaces., CRM Proc. and Lecture Notes, Montréal, 56 (2013), 55-66.
[11]	T. Coulhon and A. Sikora, Gaussian heat kernel bounds via Phragmèn-Lindelöf theorem, Proc. London. Math. Soc., 96 (2008), 507-544. doi: 10.1112/plms/pdm050.
[12]	E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-334. doi: 10.2307/2374577.
[13]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92. Cambridge Univ. Press, Cambridge, 1990.
[14]	O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité, J. Funct. Anal., 111 (1993), 155-196. doi: 10.1006/jfan.1993.1008.
[15]	V. H. Nguyen, Dimensional variance estimates of Brascamp-Lieb type and a local approach to dimensional Prékopa theorem, J. Funct. Anal., 266 (2014), 931-955. doi: 10.1016/j.jfa.2013.11.003.
[16]	L. Saloff-Coste, Sobolev inequalities in familiar and unfamiliar settings, In Sobolev spaces in mathematics. I, Int. Math. Ser., Springer, New York, 8 (2009), 299-343. doi: 10.1007/978-0-387-85648-3_11.
[17]	F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.
[18]	F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.

show all references

References:

[1]	D. Bakry, F. Barthe, P. Cattiaux and A. Guillin, A simple proof of the Poincaré inequality for a large class of probability measures including the log-concave case, Elec. Comm. Prob., 13 (2008), 60-66. doi: 10.1214/ECP.v13-1352.
[2]	D. Bakry, F. Bolley, I. Gentil and P. Maheux, Weighted Nash inequalities, Revista Mat. Iberoam., 28 (2012), 879-906. doi: 10.4171/RMI/695.
[3]	D. Bakry, I. Gentil and M. Ledoux, Analysis and Geometry of Markov Diffusion Operators, Grund. math. Wiss., vol. 348. Cham, 2014. doi: 10.1007/978-3-319-00227-9.
[4]	S. G. Bobkov and M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures, Ann. Prob., 37 (2009), 403-427. doi: 10.1214/08-AOP407.
[5]	S. Boutayeb, T. Coulhon and A. Sikora, A new approach to pointwise heat kernel upper bounds on doubling metric measure spaces, Preprint, november 2013.
[6]	E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. H. Poincaré Prob. Stat., 23 (1987), 245-287.
[7]	P. Cattiaux, N. Gozlan, A. Guillin and C. Roberto, Functional inequalities for heavy tailed distributions and application to isoperimetry, Elec. J. Prob., 15 (2010), 346-385. doi: 10.1214/EJP.v15-754.
[8]	P. Cattiaux, A. Guillin, F.-Y. Wang and L. Wu, Lyapunov conditions for super Poincaré inequalities, J. Funct. Anal., 256 (2009), 1821-1841. doi: 10.1016/j.jfa.2009.01.003.
[9]	T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539. doi: 10.1006/jfan.1996.0140.
[10]	T. Coulhon, Heat kernel estimates, Sobolev-type inequalities and Riesz transform on noncompact Riemannian manifolds, In Analysis and geometry of metric measure spaces., CRM Proc. and Lecture Notes, Montréal, 56 (2013), 55-66.
[11]	T. Coulhon and A. Sikora, Gaussian heat kernel bounds via Phragmèn-Lindelöf theorem, Proc. London. Math. Soc., 96 (2008), 507-544. doi: 10.1112/plms/pdm050.
[12]	E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-334. doi: 10.2307/2374577.
[13]	E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92. Cambridge Univ. Press, Cambridge, 1990.
[14]	O. Kavian, G. Kerkyacharian and B. Roynette, Quelques remarques sur l'ultracontractivité, J. Funct. Anal., 111 (1993), 155-196. doi: 10.1006/jfan.1993.1008.
[15]	V. H. Nguyen, Dimensional variance estimates of Brascamp-Lieb type and a local approach to dimensional Prékopa theorem, J. Funct. Anal., 266 (2014), 931-955. doi: 10.1016/j.jfa.2013.11.003.
[16]	L. Saloff-Coste, Sobolev inequalities in familiar and unfamiliar settings, In Sobolev spaces in mathematics. I, Int. Math. Ser., Springer, New York, 8 (2009), 299-343. doi: 10.1007/978-0-387-85648-3_11.
[17]	F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.
[18]	F.-Y. Wang, Functional inequalities and spectrum estimates: The infinite measure case, J. Funct. Anal., 194 (2002), 288-310. doi: 10.1006/jfan.2002.3968.

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