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Non ultracontractive heat kernel bounds by Lyapunov conditions
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 |
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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