zbMATH — the first resource for mathematics

Heat kernels on metric spaces and a characterisation of constant functions. (English) Zbl 1062.60076
The main result states that the only \(u\in L^ {2}(X, m)\) for which \[ \int _ {X} \int _ {X} \frac{(u(x)-u(y))^ {2}}{\rho (x, y)^ {d+d_ {w}}} \,dm(x) \,dm(y) <+\infty \] are the constants. Here \((X, \rho)\) is a locally compact metric space and \(m\) is a Borel measure on \(X\), for which the volume of the balls of radius \(r\) behaves as \(r^ {d}\). The proof uses the Dirichlet forms theory, hence the existence of a strongly continuous semigroup of contractions on \(L^ {2}(X, m)\), whose transition density satisfies some two-sided Gaussian estimate, is assumed. Examples are included, showing the relation with recent results of J. Bourgain, H. Brézis, Y. Li, P. Mironescu, L. Nirenberg; see H. Brézis [Russ. Math. Surv. 57, No. 4, 693–708 (2002); translation from Usp. Mat. Nauk 57, No. 4, 59–74 (2002; Zbl 1072.46020)] for a detailed account of these results.

60J35 Transition functions, generators and resolvents
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
Full Text: DOI
[1] Barlow, M.T.: Diffusion on fractals, In: Lectures on Probability and Statistics, Ecole d’Eté de Prob. de St. Flour XXV – 1995, Lecture Notes in Mathematics no. 1690, Springer-Verlag, Berlin 1998 · Zbl 0916.60069
[2] Barlow, M.T., Bass, R.F.: Brownian motion and harmonic analysis on Sierpiński carpets. Canadian J. Math. 51 (4), 673–744 (1999) · Zbl 0945.60071 · doi:10.4153/CJM-1999-031-4
[3] Bass, R.F.: Probabilistic techniques in analysis, Prob. and its Appl.,Springer-Verlag, New York 1995
[4] Bourdon, M., Pajot, H.: Cohomologie p et espaces de Besov. J. reine angew. Math. 558, 85–108 (2003) · Zbl 1044.20026 · doi:10.1515/crll.2003.043
[5] Bourgain, J., Brézis, H., Mironescu, P.: Lifting in Sobolev spaces. J. Anal. Math. 80 (2), 37–86 (2000) · Zbl 0967.46026 · doi:10.1007/BF02791533
[6] Bourgain, J., Brézis, H., Mironescu, P.: Another look at Sobolev spaces. In: Menaldi, J.L. et al. (eds.), Optimal Control and PDE. In honour of Prof. A. Bensoussan 60th Birthday, Amstredam, IOS Press 2001 · Zbl 1103.46310
[7] Bourgain, J., Brézis, H., Mironescu, P.: Limiting embedding theorems for Ws,p when s 1 and applications. J. Anal. Math. 87, 77–101 (2002) · Zbl 1029.46030 · doi:10.1007/BF02868470
[8] Brézis, H.: How to recognize constant functions (Russian), Uspekhi Mat. Nauk, 57 (4346), 59–74 (2002); translation in: Russian Math. Surveys, 57 (4), 693–708 (2002)
[9] Brézis, H., Li, Y., Mironescu, P., Nirenberg, L.: Degree and Sobolev spaces. Topol. Methods Nonl. Anal., 13, 181–190 (1999) · Zbl 0956.46024
[10] Carlen, E.A., Kusuoka, S., Stroock, D.W.: Upper bounds for symmetric Markov transition functions. Ann. IHP 23, 245–287 (1987) · Zbl 0634.60066
[11] Fukushima, M.: Dirichlet Forms and Markov Processes. Kodansha-North Holland, 1980 · Zbl 0422.31007
[12] Grigoryan, A.: Heat kernels and function theory on metric measure spaces, Contemp. Math. 338, 143–172 (2003) · Zbl 1048.58021
[13] Grigoryan, A.: Heat kernel bounds on fractal spaces. Preprint (2003)
[14] Grigoryan, A., Hu, J., Lau, K.S.: Heat kernels on metric measure spaces and an application to semilinear elliptic equations. Trans. Amer. Math. Soc. 355, 2065–2095 (2003) · Zbl 1018.60075 · doi:10.1090/S0002-9947-03-03211-2
[15] Hajłasz, P.: Personal communication
[16] Hambly, B.M., Kumagai, T.: Transition density estimates for diffusion processes on post critically finite self-similar fractals. Proc. London. Math. Soc. 78, 431–458 (1999) · Zbl 1027.60087 · doi:10.1112/S0024611599001744
[17] Jonsson, A.: Brownian motion on fractals and function spaces. Math. Z. 222, 495–504 (1996) · Zbl 0863.60079
[18] Koskela, P., Rajala, K., Shanmugalingam, N.: Lipschitz continuity of Cheeger- harmonic functions in metric measure spaces. J. Funct. Anal. 202, 147–173 (2003) · Zbl 1027.31006 · doi:10.1016/S0022-1236(02)00090-3
[19] Kumagai, T.: Estimates on transition densities for Brownian motion on nested fractals, Prob. Th. Rel. Fields, 96, 205–224 (1993) · Zbl 0792.60073 · doi:10.1007/BF01192133
[20] Kumagai, T., Sturm, K.-T.: Construction of diffusion processes on fractals, d–sets and general metric measure spaces. Preprint (2003)
[21] Lindstrøm, T.: Brownian motion on nested fractals. Mem. Amer. Math. Soc. 420, 1–128 (1990) · Zbl 0688.60065
[22] Pietruska-Pałuba, K.: On function spaces related to fractional diffusions on d–sets. Stoch. Stoch. Rep. 70, 153–164 (2000) · Zbl 0967.60079
[23] Stós, A.: Symmetric \(\alpha\)–stable processes on d–sets. Bull. Polish Acad. Sci. Math. 48 (3), 237–245 (2000) · Zbl 0984.60087
[24] Sturm, K.-T.: Diffusion processes and heat kernels on metric spaces. Ann. Probab. 26, 1–55 (1998) · Zbl 0936.60074 · doi:10.1214/aop/1022855410
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.