American Institute of Mathematical Sciences

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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

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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539. doi: 10.1006/jfan.1996.0140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-334. doi: 10.2307/2374577.  Google Scholar

[13]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92. Cambridge Univ. Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

T. Coulhon, Ultracontractivity and Nash type inequalities, J. Funct. Anal., 141 (1996), 510-539. doi: 10.1006/jfan.1996.0140.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

E. B. Davies, Explicit constants for Gaussian upper bounds on heat kernels, Amer. J. Math., 109 (1987), 319-334. doi: 10.2307/2374577.  Google Scholar

[13]

E. B. Davies, Heat Kernels and Spectral Theory, Cambridge Tracts in Math., vol. 92. Cambridge Univ. Press, Cambridge, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

F.-Y. Wang, Functional inequalities for empty essential spectrum, J. Funct. Anal., 170 (2000), 219-245. doi: 10.1006/jfan.1999.3516.  Google Scholar

[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.  Google Scholar

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