Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Smooth bumps, a Borel theorem and partitions of smooth functions on p.c.f. fractals

Authors: Luke G. Rogers, Robert S. Strichartz and Alexander Teplyaev
Journal: Trans. Amer. Math. Soc. 361 (2009), 1765-1790
MSC (2000): Primary 28A80; Secondary 31C45, 60J60
Published electronically: November 24, 2008
MathSciNet review: 2465816
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We provide two methods for constructing smooth bump functions and for smoothly cutting off smooth functions on fractals, one using a probabilistic approach and sub-Gaussian estimates for the heat operator, and the other using the analytic theory for p.c.f. fractals and a fixed point argument. The heat semigroup (probabilistic) method is applicable to a more general class of metric measure spaces with Laplacian, including certain infinitely ramified fractals; however the cutoff technique involves some loss in smoothness. From the analytic approach we establish a Borel theorem for p.c.f. fractals, showing that to any prescribed jet at a junction point there is a smooth function with that jet. As a consequence we prove that on p.c.f. fractals smooth functions may be cut off with no loss of smoothness, and thus can be smoothly decomposed subordinate to an open cover. The latter result provides a replacement for classical partition of unity arguments in the p.c.f. fractal setting.

References [Enhancements On Off] (What's this?)

  • 1. Martin T. Barlow, Diffusions on fractals, Lectures on probability theory and statistics (Saint-Flour, 1995), Lecture Notes in Math., vol. 1690, Springer, Berlin, 1998, pp. 1-121. MR 1668115 (2000a:60148)
  • 2. Martin T. Barlow and Richard F. Bass, The construction of Brownian motion on the Sierpiński carpet, Ann. Inst. H. Poincaré Probab. Statist. 25 (1989), no. 3, 225-257. MR 1023950 (91d:60183)
  • 3. -, Brownian motion and harmonic analysis on Sierpinski carpets, Canad. J. Math. 51 (1999), no. 4, 673-744. MR 1701339 (2000i:60083)
  • 4. Oren Ben-Bassat, Robert S. Strichartz, and Alexander Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999), no. 2, 197-217. MR 1707752 (2001e:31016)
  • 5. Nitsan Ben-Gal, Abby Shaw-Krauss, Robert S. Strichartz, and Clint Young, Calculus on the Sierpinski gasket II: Point singularities, eigenfunctions, and normal derivatives of the heat kernel, Trans. Amer. Math. Soc. 358 (2006), 3883-3936. MR 2219003 (2007h:28009)
  • 6. E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239 (90e:35123)
  • 7. 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)
  • 8. Masatoshi Fukushima, Yōichi 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 (96f:60126)
  • 9. A. A. Grigor$ '$yan, The heat equation on noncompact Riemannian manifolds, Mat. Sb. 182 (1991), no. 1, 55-87. MR 1098839 (92h:58189)
  • 10. Alexander Grigor$ '$yan and Andras Telcs, Sub-Gaussian estimates of heat kernels on infinite graphs, Duke Math. J. 109 (2001), no. 3, 451-510. MR 1853353 (2003a:35085)
  • 11. Alexander Grigor$ '$yan and András Telcs, Harnack inequalities and sub-Gaussian estimates for random walks, Math. Ann. 324 (2002), no. 3, 521-556. MR 1938457 (2003i:58068)
  • 12. 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)
  • 13. P. Edward Herman, Roberto Peirone, and Robert S. Strichartz, $ p$-energy and $ p$-harmonic functions on Sierpinski gasket type fractals, Potential Anal. 20 (2004), no. 2, 125-148. MR 2032945 (2004k:31023)
  • 14. Jun Kigami, Analysis on fractals, Cambridge Tracts in Mathematics, vol. 143, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)
  • 15. -, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), no. 2, 399-444. MR 2017320 (2004m:31010)
  • 16. Takashi Kumagai, Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals, Publ. Res. Inst. Math. Sci. 33 (1997), no. 2, 223-240. MR 1442498 (98k:60130)
  • 17. Tom Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128. MR 988082 (90k:60157)
  • 18. Robert Meyers, Robert S. Strichartz, and Alexander Teplyaev, Dirichlet forms on the Sierpiński gasket, Pacific J. Math. 217 (2004), no. 1, 149-174. MR 2105771 (2005k:31028)
  • 19. 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
  • 20. Kasso A. Okoudjou, Luke G. Rogers, and Robert S. Strichartz, Generalized eigenfunctions and a Borel theorem on the Sierpinski gasket, To appear in Canad. Math. Bull.
  • 21. Roberto Peirone, Convergence and uniqueness problems for Dirichlet forms on fractals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), no. 2, 431-460. MR 1769995 (2001i:31016)
  • 22. Luke G. Rogers and Robert S. Strichartz, Distributions on p.c.f. fractafolds, in preparation.
  • 23. C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. École Norm. Sup. (4) 30 (1997), no. 5, 605-673. MR 1474807 (98h:60118)
  • 24. L. Saloff-Coste, A note on Poincaré, Sobolev, and Harnack inequalities, Internat. Math. Res. Notices (1992), no. 2, 27-38. MR 1150597 (93d:58158)
  • 25. Robert S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc. 357 (2005), no. 2, 571-615 (electronic). MR 2095624 (2005m:31016)
  • 26. Robert S. Strichartz, Differential equations on fractals: A tutorial, Princeton University Press, 2006. MR 2246975 (2007f:35003)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2000): 28A80, 31C45, 60J60

Retrieve articles in all journals with MSC (2000): 28A80, 31C45, 60J60

Additional Information

Luke G. Rogers
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Robert S. Strichartz
Affiliation: Department of Mathematics, Cornell University, Malott Hall, Ithaca, New York 14853-4201

Alexander Teplyaev
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009

Received by editor(s): January 4, 2007
Published electronically: November 24, 2008
Additional Notes: This research was supported in part by the National Science Foundation, Grant DMS-0140194
This research was supported in part by the National Science Foundation, Grant DMS-0505622
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society