Nash-type inequalities and heat kernels for non-local Dirichlet forms

Kyushu Journal of Mathematics

Volumn 60, Issue 2, 2007, Pages 245-265

  Hu, Jiaxin ^a   Kumagai, Takashi ^b  

^a TSINGHUA UNIVERSITY   (China)

^b KYOTO UNIVERSITY   (Japan)

Author keywords

Dirichlet forms; Fractals; Heat kernels; Jump processes; L vy system; Nash's inequality; Parabolic Harnack inequality

Indexed keywords

EID: 34547107441     PISSN: 13406116     EISSN: None     Source Type: Journal    
DOI: 10.2206/kyushujm.60.245     Document Type: Article

References (21)

1. D. Bakry, T. Coulhon, M. Ledoux and L. Saloff-Coste. Sobolev inequalities in disguise, Indiana Univ. Math. J. 44 (1995), 1033-1074.
2. M. T. Barlow. Diffusions on fractals. Lectures on Probability Theory and Statistics (Lecture Notes in Mathematics, 1690). Springer, Berlin, 1998, pp. 1-121.
3. M. T. Barlow and R. F. Bass. Brownian motion and harmonic analysis on Sierpínski carpets. Canad. J. Math. 51 (1999), 673-744.
4. R. F. Bass and D. A. Levin. Transition probabilities for symmetric jump processes. Trans. Amer. Math. Soc. 354 (2002), 2933-2953.
5. R. F. Bass and D. A. Levin. Harnack inequalities for jump processes. Potential Anal. 17 (2002), 375-388.
6. K. Bogdan, A. Stós and P. Sztonyk. Harnack inequality for stable processes on d-sets. Studia Math. 158 (2003), 163-198.
7. E. A. Carlen, S. Kusuoka and D. W. Stroock. Upper bounds for symmetric Markov transition functions. Ann. Inst. Heri. Poincaré-Probab. Statist. 23 (1987), 245-287.
8. Z. Q. Chen and T. Kumagai. Heat kernel estimates for stable-like processes on d-sets. Stochastic Process Appl. 108 (2003), 27-62.
9. Z. Q. Chen and T. Kumagai. Heat kernel estimates for jump processes of mixed types on metric measure spaces. In preparation.
10. M. Fukushima, Y. Oshima and M. Takeda. Dirichlet Forms and Symmetric Markov Processes. de Gruyter, Berlin, 1994.
11. M. Fukushima and T. Uemura. On Sobolev and capacitary inequalities for contractive Besov spaces over d-sets. Potential Anal. 18 (2003), 59-77.
12. A. Grigor'yan. Heat kernels and function theory on metric measure spaces. Heat kernels and analysis on manifolds, graphs, and metric spaces (Contemp. Math., 338). Ed. P. Auscher et al. American Mathematical Society, Providence, RI, 2003, pp. 143-172.
13. A. Grigor'yan, J. Hu and K.-S. Lau. Heat kernels on metric-measure spaces and an application to semilinear elliptic equations. Trans. Amer. Math. Soc. 355 (2003), 2065-2095.
14. B. M. Hambly and T. Kumagai. Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London Math. Soc. 78 (1999), 431-458.
15. J. Hu. A note on Hajłasz-Sobolev spaces on fractals. J. Math. Anal. Appl. 280 (2003), 91-101.
16. J. Hu and M. Zähble. Jump processes and nonlinear fractional heat equations on fractals. Math. Nachr. 279 (2006), 150-163.
17. T. Kumagai. Some remarks on stable-like jump processes on fractals. Trends in Mathematics, Fractals (Graz, 2001). Eds. P. Grabner and W. Woess. Birkhäuser, Basel, 2002, pp. 185-196.
18. T. Kumagai. Function spaces and stochastic processes on fractals. Fractal Geometry and Stochastics III (Progr. Probab., 57). Ed. C. Bandt et al. Birkhäuser, Basel, 2004, pp. 221-234.
19. R. S. Phillips. Perturbation theory for semi-groups of linear operators. Trans. Amer. Math. Soc. 74 (1953), 199-221.
20. L. Saloff-Coste. A note on Poincaré, Sobolev, and Harnack inequalities. Internat. Math. Res. Notices 2 (1992), 27-38.
21. A. Stós. Symmetric α-stable processes on d-sets. Bull. Pol. Acad. Sci. Math. 48 (2000), 237-245.