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Supposing that the metric space in question supports a fractional diffusion, we prove that after introducing an appropriate multiplicative factor, the Gagliardo seminorms ||ƒ||wσ,2 of a function ƒ ∈ L²(E, μ) have the property [wzór...] where ε is the Dirichlet form relative to the fractional diffusion.
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Rocznik
Tom
Strony
257--266
Opis fizyczny
Bibliogr. 22 poz.
Twórcy
autor
- Institute of Mathematics, University of Warsaw, Banacha 2 02-097 Warszawa, Poland, kpp@mimuw.edu.pl
Bibliografia
- [1] R. A. Adams, Sobolev Spaces, Academic Press, New York, 1975.
- [2] M. T. Barlow, Diffusion on fractals, in: Lectures on Probability and Statistics, École d’Été de Probabilités de St. Flour XXV-1995, Lecture Notes in Math. 1690, Springer, New York, 1998, 1-121.
- [3] M. T. Barlow and R. F. Bass, Brownian motion and analysis on Sierpiński carpets, Canad. J. Math. 51 (1999), 673-744.
- [4] R. M. Blumenthal and R. K. Getoor, Markov Processes and Potential Theory, Pure Appl. Math., Academic Press, New York, 1968.
- [5] K. Bogdan, A. Stós and P. Sztonyk, Harnack inequality for stable processes on d-sets, Studia Math. 158 (2003), 163-198.
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- [7] —, —, —, Limiting embedding theorems for Ws,p when s ↑ 1 and applications, J. Anal. Math. 87 (2002), 77-101.
- [8] H. Brézis, How to recognize constant functions, Uspekhi Mat. Nauk, 57 (2002), no. 4, 59-74 (in Russian); English transl.: Russian Math. Surveys 57 (2002), 693-708.
- [9] E. A. Carlen, S. Kusuoka and D. W. Stroock, Upper bounds for symmetric Markov transition functions, Ann. Inst. Poincare Probab. Statist. 23 (1987), 245-287.
- [10] G. Doetsch, Introduction to the Theory and Application of the Laplace Transformation, Springer, New York, 1974.
- [11] M. Fukushima, Dirichlet Forms and Markov Processes, Kodansha-North-Holland, 1980.
- [12] A. Grigoryan, Heat kernels and function theory on metric measure spaces, in: Contemp. Math. 338, Amer. Math. Soc., 2003, 143-172.
- [13] A. Grigoryan, 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] A. Grigoryan and T. Kumagai, On the dichotomy of the heat kernel two sided-estimates, in: Proc. Sympos. Pure Math. 77, Amer. Math. Soc., 2008, 199-210.
- [15] 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.
- [16] A. Jonsson, Brownian motion on fractals and function spaces, Math. Z. 222 (1996), 495-504.
- [17] T. Kumagai, Function spaces and stochastic processes on fractals, in: Fractal Geometry and Stochastics III, C. Bandt et al. (eds.), Progr. Probab. 57, Birkhäuser, 2004, 221-234.
- [18] W. Masja [V. Maz’ya] und J. Nagel, Über äquivalente Normierung der anisotropen Funktionalräume Hμ(Rn), Beiträge Anal. 12 (1978), 7-17.
- [19] K. Pietruska-Pałuba, On function spaces related to fractional diffusions on d-sets, Stoch. Stoch. Rep. 70 (2000), 153-164.
- [20] —, Heat kernels on metric spaces and a characterization of constant functions, Manuscripta Math. 115 (2004), 389-399.
- [21] A. Stós, Symmetric α-stable processes on d-sets, Bull. Polish Acad. Sci. Math. 48 (2000), 237-245.
- [22] K. T. Sturm, Diffusion processes and heat kernels on metric spaces, Ann. Probab. 26 (1998), 1-55.
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Bibliografia
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bwmeta1.element.baztech-article-BAT5-0035-0008