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Estimates for the resolvent kernel of the Laplacian on p.c.f. self-similar fractals and blowups

Author: Luke G. Rogers
Journal: Trans. Amer. Math. Soc. 364 (2012), 1633-1685
MSC (2010): Primary 28A80, 60J35
Published electronically: October 24, 2011
MathSciNet review: 2869187
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A central issue in studying resistance forms on post-critically finite self-similar sets is the behavior of the resolvent of the Laplacian operator. The kernel of this operator may be obtained by a self-similar series and estimated on the right half of the complex plane via probabilistic bounds for the associated heat kernel. This paper generalizes the known upper estimates to the complement of the negative real axis in the complex plane via a new method using resistance forms and the Phragmen-Lindelöf theorem. Consequences include a proof of the self-similar structure of the resolvent on blow-ups, estimates for other spectral operators, and a new proof of the known upper estimates for the heat kernel on these sets.

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  • 1. Martin T. Barlow and Edwin A. Perkins, Brownian motion on the Sierpiński gasket, Probab. Theory Related Fields 79 (1988), no. 4, 543-623. MR 966175 (89g:60241)
  • 2. Thierry Coulhon and Adam Sikora, Gaussian heat kernel upper bounds via the Phragmén-Lindelöf theorem, Proc. Lond. Math. Soc. (3) 96 (2008), no. 2, 507-544. MR 2396848
  • 3. Pat J. Fitzsimmons, Ben M. Hambly, and Takashi Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys. 165 (1994), no. 3, 595-620. MR 1301625 (95j:60122)
  • 4. B. M. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite self-similar fractals, Proc. London Math. Soc. (3) 78 (1999), no. 2, 431-458. MR 1665249 (99m:60118)
  • 5. John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713-747. MR 625600 (82h:49026)
  • 6. Marius Ionescu, Erin P. J. Pearse, Luke G. Rogers, Huo-Jun Ruan, and Robert S. Strichartz, The resolvent kernel for PCF self-similar fractals, Trans. Amer. Math. Soc. 362 (2010), no. 8, 4451-4479. MR 2608413
  • 7. Marius Ionescu and Luke G. Rogers, Complex powers of the laplacian on affine nested fractals as Calderón-Zygmund operators., Preprint 2010.
  • 8. Jun Kigami, Harmonic calculus on p.c.f. self-similar sets, Trans. Amer. Math. Soc. 335 (1993), no. 2, 721-755. MR 1076617 (93d:39008)
  • 9. -, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • 10. -, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), no. 2, 399-444. MR 2017320 (2004m:31010)
  • 11. -, Volume doubling measures and heat kernel estimates on self-similar sets, Mem. Amer. Math. Soc. 199 (2009), no. 932, viii+94. MR 2512802
  • 12. 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 (94m:58225)
  • 13. Takashi Kumagai, Estimates of transition densities for Brownian motion on nested fractals, Probab. Theory Related Fields 96 (1993), no. 2, 205-224. MR 1227032 (94e:60068)
  • 14. B. Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, vol. 150, American Mathematical Society, Providence, RI, 1996. In collaboration with and with a preface by Yu. Lyubarskii, M. Sodin and V. Tkachenko. Translated from the Russian manuscript by Tkachenko. MR 1400006 (97j:30001)
  • 15. Jonathan Needleman, Robert S. Strichartz, Alexander Teplyaev, and Po-Lam Yung, Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series, J. Funct. Anal. 215 (2004), no. 2, 290-340. MR 2150975 (2006h:28013)
  • 16. Luke G. Rogers and Robert S. Strichartz, Distributions on p.c.f. fractafolds, Journal d'Analyse Mathématique 112 (2011), no. 1, 137-191.
  • 17. Robert S. Strichartz, Fractals in the large, Canad. J. Math. 50 (1998), no. 3, 638-657. MR 1629847 (99f:28015)
  • 18. -, Analysis on products of fractals, Trans. Amer. Math. Soc. 357 (2005), no. 2, 571-615 (electronic). MR 2095624 (2005m:31016)
  • 19. -, Differential equations on fractals, Princeton University Press, Princeton, NJ, 2006, A tutorial. MR 2246975 (2007f:35003)
  • 20. Alexander Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal. 159 (1998), no. 2, 537-567. MR 1658094 (99j:35153)

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Additional Information

Luke G. Rogers
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269

Received by editor(s): September 1, 2010
Received by editor(s) in revised form: January 27, 2011
Published electronically: October 24, 2011
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.