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Distribution theory on P.C.F. fractals

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Abstract

We construct a theory of distributions in the setting of analysis on post-critically finite self-similar fractals, and on fractafolds and products based on such fractals. The results include basic properties of test functions and distributions, a structure theorem showing that distributions are locally-finite sums of powers of the Laplacian applied to continuous functions, and an analysis of the distributions with point support. Possible future applications to the study of hypoelliptic partial differential operators are suggested.

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References

  1. C. Avenancio-Leon and R. S. Strichartz, Local behavior of harmonic functions on the Sierpinski gasket, Illinois J. Math. 51 (2007), 1061–1075. MR 2417415

    MATH  MathSciNet  Google Scholar 

  2. M. T. Barlow and J. Kigami, Localized eigenfunctions of the Laplacian on p.c.f. self-similar sets, J. London Math. Soc. (2) 56 (1997), 320–332. MR 1489140 (99b:35162)

    Article  MATH  MathSciNet  Google Scholar 

  3. O. Ben-Bassat, R. S. Strichartz, and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. Funct. Anal. 166 (1999), 197–217. MR 1707752 (2001e:31016)

    Article  MATH  MathSciNet  Google Scholar 

  4. N. Ben-Gal, A. Shaw-Krauss, R. S. Strichartz, and C. 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 (electronic). MR 2219003 (2007h:28009)

    Article  MATH  MathSciNet  Google Scholar 

  5. N. Ben-Gal, A. Shaw-Krauss, R. S. Strichartz, and C. 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 (electronic). MR 2219003

    Article  MATH  MathSciNet  Google Scholar 

  6. B. Bockelman and R. S. Strichartz, Partial differential equations on products of Sierpinski gaskets, Indiana Univ. Math. J. 56 (2007), 1361–1375. MR 2333476 (2008h:31012)

    Article  MATH  MathSciNet  Google Scholar 

  7. P. J. Fitzsimmons, B. M. Hambly, and T. Kumagai, Transition density estimates for Brownian motion on affine nested fractals, Comm. Math. Phys. 165 (1994), 595–620. MR 1301625 (95j:60122)

    Article  MATH  MathSciNet  Google Scholar 

  8. H. Furstenberg and H. Kesten, Products of random matrices, Ann. Math. Statist 31 (1960), 457–469. MR 0121828 (22 #12558)

    Article  MATH  MathSciNet  Google Scholar 

  9. 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), 431–458. MR 1665249 (99m:60118)

    Article  MATH  MathSciNet  Google Scholar 

  10. B.M. Hambly, V. Metz, and A. Teplyaev, Self-similar energies on post-critically finite self-similar fractals, J. London Math. Soc. (2) 74 (2006), 93–112. MR 2254554 (2007i:31011)

    Article  MATH  MathSciNet  Google Scholar 

  11. L. Hörmander, The Analysis of Linear Partial Differential Operators. II Differential Operators with Constant Coefficients, Springer-Verlag, Berlin, 1983. MR 705278 (85g:35002b)

    MATH  Google Scholar 

  12. M. Ionescu, E. P. J. Pearse, Huo-Jun Ruan, L. G. Rogers, and Robert S. Strichartz, The resolvent kernel for pcf self-similar fractals, Trans. Amer. Math. Soc. 362 (2010), 4451–4479. MR 2608413

    Article  MATH  MathSciNet  Google Scholar 

  13. J. Kigami, Analysis on Fractals, Cambridge University Press, Cambridge, 2001. MR 1840042 (2002c:28015)

    Book  MATH  Google Scholar 

  14. J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), 399–444. MR 2017320 (2004m:31010)

    Article  MATH  MathSciNet  Google Scholar 

  15. T. Kumagai, Short time asymptotic behaviour and large deviation for Brownian motion on some affine nested fractals, Publ. Res. Inst. Math. Sci. 33 (1997), 223–240. MR 1442498 (98k:60130)

    Article  MATH  MathSciNet  Google Scholar 

  16. S. Kusuoka, Dirichlet forms on fractals and products of random matrices, Publ. Res. Inst. Math. Sci. 25 (1989), 659–680. MR 1025071 (91m:60142)

    Article  MATH  MathSciNet  Google Scholar 

  17. T. Lindstrøm, Brownian motion on nested fractals, Mem. Amer. Math. Soc. 83 (1990), no. 420, iv+128 pp. MR 988082 (90k:60157)

  18. R. Meyers, R. S. Strichartz, and A. Teplyaev, Dirichlet forms on the Sierpiński gasket, Pacific J. Math. 217 (2004), 149–174. MR 2105771 (2005k:31028)

    Article  MATH  MathSciNet  Google Scholar 

  19. J. Needleman, R. S. Strichartz, A. Teplyaev, and Po-Lam Yung, Calculus on the Sierpinski gasket. I. Polynomials, exponentials and power series, J. Funct. Anal. 215 (2004), 290–340. MR 2150975

    Article  MATH  MathSciNet  Google Scholar 

  20. K. A. Okoudjou, L. G. Rogers, and R. S. Strichartz, Generalized eigenfunctions and a Borel theorem on the Sierpinski gasket, Canad. Math. Bull. 52 (2009), 105–116.

    Article  MATH  MathSciNet  Google Scholar 

  21. R. Peirone, Convergence and uniqueness problems for Dirichlet forms on fractals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), 431–460. MR 1769995 (2001i:31016)

    MATH  MathSciNet  Google Scholar 

  22. R. Peirone, Convergence of Dirichlet forms on fractals, in Topics on Concentration Phenomena and Problems with Multiple Scales, Lecture Notes Unione Mat. Ital., Vol. 2, Springer, Berlin, 2006, pp. 139–188. MR 2267882 (2008b:28013)

    Chapter  Google Scholar 

  23. R. Peirone, Existence of eigenforms on fractals with three vertices, Proc. Roy. Soc. Edinburgh Sect. A 137 (2007), 1073–1080. MR 2359927

    Article  MATH  MathSciNet  Google Scholar 

  24. R. Peirone, Uniqueness of eigenforms on nicely separated fractals, in Analysis on Graphs and its Applications, Amer. Math. Soc., Providence, RI, 2008, pp. 231–241.

    Google Scholar 

  25. A. Pelander, Solvability of differential equations on open subsets of the Sierpiński gasket, J. Analy. Math. 102 (2007), 359–369. MR 2346562 (2009d:31020)

    Article  MATH  MathSciNet  Google Scholar 

  26. A. Pelander and A. Teplyaev, Infinite dimensional i.f.s. and smooth functions on the Sierpiński gasket, Indiana Univ. Math. J. 56 (2007), 1377–1404. MR 2333477

    Article  MATH  MathSciNet  Google Scholar 

  27. A. Pelander and A. Teplyaev, Products of random matrices and derivatives on p.c.f. fractals, J. Funct. Anal. 254 (2008), 1188–1216. MR 2386935

    Article  MATH  MathSciNet  Google Scholar 

  28. L. G. Rogers, R. S. Strichartz, and A. Teplyaev, Smooth bumps, a Borel theorem and partitions of smooth functions on P.C.F. fractals, Trans. Amer. Math. Soc. 361 (2009), 1765–1790. MR 2465816

    Article  MATH  MathSciNet  Google Scholar 

  29. W. Rudin, Functional Analysis, second edition, McGraw-Hill Inc., New York, 1991. MR 1157815 (92k:46001)

    MATH  Google Scholar 

  30. C. Sabot, Existence and uniqueness of diffusions on finitely ramified self-similar fractals, Ann. Sci. École Norm. Sup. (4) 30 (1997), 605–673. MR 1474807 (98h:60118)

    MATH  MathSciNet  Google Scholar 

  31. A. Sikora, Multivariable spectral multipliers and quasielliptic operators, Indiana Univ. Math. J. 58 (2009), 317–334. MR 2504414 (2010d:47068)

    Article  MATH  MathSciNet  Google Scholar 

  32. R. S. Strichartz, Taylor approximations on Sierpinski gasket type fractals, J. Funct. Anal. 174 (2000), 76–127. MR 1761364 (2001i:31018)

    Article  MATH  MathSciNet  Google Scholar 

  33. R. S. Strichartz, Fractafolds based on the Sierpiński gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 4019–4043 (electronic). MR 1990573 (2004b:28013)

    Article  MATH  MathSciNet  Google Scholar 

  34. R. S. Strichartz, Function spaces on fractals, J. Funct. Anal. 198 (2003), 43–83. MR 1962353 (2003m:46058)

    Article  MATH  MathSciNet  Google Scholar 

  35. R. S. Strichartz, Analysis on products of fractals, Trans. Amer. Math. Soc. 357 (2005), 571–615 (electronic). MR 2095624 (2005m:31016)

    Article  MATH  MathSciNet  Google Scholar 

  36. R. S. Strichartz, Solvability for differential equations on fractals, J. Anal. Math. 96 (2005), 247–267. MR 2177187 (2006j:35092)

    Article  MATH  MathSciNet  Google Scholar 

  37. R. S. Strichartz, Differential Equations on Fractals, Princeton University Press, Princeton, NJ, 2006. MR 2246975 (2007f:35003)

    MATH  Google Scholar 

  38. A. Teplyaev, Spectral analysis on infinite Sierpiński gaskets, J. Funct. Anal. 159 (1998), 537–567. MR 1658094 (99j:35153)

    Article  MATH  MathSciNet  Google Scholar 

  39. A. Teplyaev, Gradients on fractals, J. Funct. Anal. 174 (2000), 128–154. MR 1761365 (2001h:31012)

    Article  MATH  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics, University of Connecticut, Storrs, CT, 06269-3009, USA

    Luke G. Rogers

  2. Department of Mathematics, Cornell University, Ithaca, NY, 14850-4201, USA

    Robert S. Strichartz

Authors
  1. Luke G. Rogers
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  2. Robert S. Strichartz
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Correspondence to Luke G. Rogers.

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Second author supported in part by NSF grant DMS 0652440.

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Rogers, L.G., Strichartz, R.S. Distribution theory on P.C.F. fractals. JAMA 112, 137–191 (2010). https://doi.org/10.1007/s11854-010-0027-y

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  • DOI: https://doi.org/10.1007/s11854-010-0027-y

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