Harmonic calculus on fractals—A measure geometric approach II

Zähle, M.

doi:10.1090/S0002-9947-05-03854-7

Harmonic calculus on fractals—A measure geometric approach II
HTML articles powered by AMS MathViewer

by M. Zähle PDF

Trans. Amer. Math. Soc. 357 (2005), 3407-3423 Request permission

Abstract:

Riesz potentials of fractal measures $\mu$ in metric spaces and their inverses are introduced. They define self–adjoint operators in the Hilbert space $L_2(\mu )$ and the former are shown to be compact. In the Euclidean case the corresponding spectral asymptotics are derived with Besov space methods. The inverses of the Riesz potentials are fractal pseudodifferential operators. For the order two operator the spectral dimension coincides with the Hausdorff dimension of the underlying fractal.

References

David R. Adams and Lars Inge Hedberg, Function spaces and potential theory, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 314, Springer-Verlag, Berlin, 1996. MR 1411441, DOI 10.1007/978-3-662-03282-4
Manfred Denker and Hiroshi Sato, Sierpiński gasket as a Martin boundary. I. Martin kernels, Potential Anal. 14 (2001), no. 3, 211–232. MR 1822915, DOI 10.1023/A:1011232724842
Gerald A. Edgar, Integral, probability, and fractal measures, Springer-Verlag, New York, 1998. MR 1484412, DOI 10.1007/978-1-4757-2958-0
K. J. Falconer, Semilinear PDEs on self-similar fractals, Comm. Math. Phys. 206 (1999), no. 1, 235–245. MR 1736985, DOI 10.1007/s002200050703
Kenneth J. Falconer and Jiaxin Hu, Non-linear elliptical equations on the Sierpiński gasket, J. Math. Anal. Appl. 240 (1999), no. 2, 552–573. MR 1731662, DOI 10.1006/jmaa.1999.6617
U. Freiberg, Maßgeometrische Laplaceoperatoren für fraktale Teilmengen der reellen Achse, Ph.D. thesis, Friedrich–Schiller University, Jena, 2000.
U. Freiberg and M. Zähle, Harmonic calculus on fractals—a measure geometric approach. I, Potential Anal. 16 (2002), no. 3, 265–277. MR 1885763, DOI 10.1023/A:1014085203265
Takahiko Fujita, A fractional dimension, self-similarity and a generalized diffusion operator, Probabilistic methods in mathematical physics (Katata/Kyoto, 1985) Academic Press, Boston, MA, 1987, pp. 83–90. MR 933819
Masatoshi Fukushima, Y\B{o}ichi Ōshima, and Masayoshi Takeda, Dirichlet forms and symmetric Markov processes, De Gruyter Studies in Mathematics, vol. 19, Walter de Gruyter & Co., Berlin, 1994. MR 1303354, DOI 10.1515/9783110889741
Alf Jonsson, Besov spaces on closed subsets of $\textbf {R}^n$, Trans. Amer. Math. Soc. 341 (1994), no. 1, 355–370. MR 1132434, DOI 10.1090/S0002-9947-1994-1132434-6
Alf Jonsson and Hans Wallin, Function spaces on subsets of $\textbf {R}^n$, Math. Rep. 2 (1984), no. 1, xiv+221. MR 820626
Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), no. 2, 721–755. MR 1076617, DOI 10.1090/S0002-9947-1993-1076617-1
Jun Kigami and Michel L. Lapidus, Weyl’s problem for the spectral distribution of Laplacians on p.c.f. self-similar fractals, Comm. Math. Phys. 158 (1993), no. 1, 93–125. MR 1243717
Ka-Sing Lau, Fractal measures and mean $p$-variations, J. Funct. Anal. 108 (1992), no. 2, 427–457. MR 1176682, DOI 10.1016/0022-1236(92)90031-D
J. Malý and U. Mosco, Remarks on measure-valued Lagrangians on homogeneous spaces, Ricerche Mat. 48 (1999), no. suppl., 217–231. Papers in memory of Ennio De Giorgi (Italian). MR 1765686
Volker Metz, Potentialtheorie auf dem Sierpiński gasket, Math. Ann. 289 (1991), no. 2, 207–237 (German). MR 1092171, DOI 10.1007/BF01446569
Umberto Mosco, Dirichlet forms and self-similarity, New directions in Dirichlet forms, AMS/IP Stud. Adv. Math., vol. 8, Amer. Math. Soc., Providence, RI, 1998, pp. 117–155. MR 1652280, DOI 10.1090/amsip/008/03
Boris Rubin, Fractional integrals and potentials, Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 82, Longman, Harlow, 1996. MR 1428214
Stefan G. Samko, Anatoly A. Kilbas, and Oleg I. Marichev, Fractional integrals and derivatives, Gordon and Breach Science Publishers, Yverdon, 1993. Theory and applications; Edited and with a foreword by S. M. Nikol′skiĭ; Translated from the 1987 Russian original; Revised by the authors. MR 1347689
Robert S. Strichartz, Self-similar measures and their Fourier transforms. I, Indiana Univ. Math. J. 39 (1990), no. 3, 797–817. MR 1078738, DOI 10.1512/iumj.1990.39.39038
Robert S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal. 174 (2000), no. 1, 76–127. MR 1761364, DOI 10.1006/jfan.2000.3580
Hans Triebel, Fractals and spectra, Monographs in Mathematics, vol. 91, Birkhäuser Verlag, Basel, 1997. Related to Fourier analysis and function spaces. MR 1484417, DOI 10.1007/978-3-0348-0034-1
Kôsaku Yosida, Functional analysis, 2nd ed., Die Grundlehren der mathematischen Wissenschaften, Band 123, Springer-Verlag New York, Inc., New York, 1968. MR 0239384
M. Zähle, Measure theoretic Laplace operators on fractals, Stochastic processes, physics and geometry: new interplays, II (Leipzig, 1999) CMS Conf. Proc., vol. 29, Amer. Math. Soc., Providence, RI, 2000, pp. 625–631. MR 1803453
M. Zähle, Riesz potentials and Liouville operators on fractals, Potential Anal. 21 (2004), no. 2, 193–208. MR 2058033, DOI 10.1023/B:POTA.0000025381.84557.a9