Analysis on products of fractals
Author:
Robert S. Strichartz
Journal:
Trans. Amer. Math. Soc. 357 (2005), 571615
MSC (2000):
Primary 31C45, 28A80
Published electronically:
September 23, 2004
MathSciNet review:
2095624
Fulltext PDF Free Access
Abstract 
References 
Similar Articles 
Additional Information
Abstract: For a class of postcritically finite (p.c.f.) fractals, which includes the Sierpinski gasket (SG), there is a satisfactory theory of analysis due to Kigami, including energy, harmonic functions and Laplacians. In particular, the Laplacian coincides with the generator of a stochastic process constructed independently by probabilistic methods. The probabilistic method is also available for nonp.c.f. fractals such as the Sierpinski carpet. In this paper we show how to extend Kigami's construction to products of p.c.f. fractals. Since the products are not themselves p.c.f., this gives the first glimpse of what the analytic theory could accomplish in the nonp.c.f. setting. There are some important differences that arise in this setting. It is no longer true that points have positive capacity, so functions of finite energy are not necessarily continuous. Also the boundary of the fractal is no longer finite, so boundary conditions need to be dealt with in a more involved manner. All in all, the theory resembles PDE theory while in the p.c.f. case it is much closer to ODE theory.
 [B]
G. Barbatis, Explicit estimates on the fundamental solution of higherorder parabolic equations with measurable coefficients, J. Diff. Eq. 174 (2001), 442463. MR 1846743 (2002i:35002)
 [BD]
G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniform operators, J. Operator Theory 36 (1996), 179198.MR 1417193 (97k:35105)
 [Ba]
M. Barlow, Diffusion on fractals, Lecture Notes Math., vol. 1690, Springer, 1998.MR 1668115 (2000a:60148)
 [BP]
M. Barlow and E. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Related Fields 79 (1988), 543623. MR 0966175 (89g:60241)
 [BST]
O. BenBassat, R. Strichartz and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. of Functional Analysis 166 (1999), 197217. MR 1707752 (2001e:31016)
 [BSSY]
N. BenGal, A. ShawKrauss, R. Strichartz and C. Young, Calculus on the Sierpinski gasket II: point singularities, eigenfunctions, and normal derivatives of the heat kernel, preprint.
 [Be]
C. Berg, Potential theory on the infinite dimensional torus, Inv. Math. 32 (1976), 49100. MR 0402093 (53:5915)
 [BrP]
J. H. Bramble and L.E. Payne, Bounds for the first derivatives of Green's function, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur (8) 42 (1967), 604610. MR 0224850 (37:449)
 [C]
T. Coulhon, Offdiagonal heat kernel lower bounds without Poincaré, J. London Math. Soc. (2) 68 (2003), 795816. MR 2010012
 [CDS]
K. Coletta, K. Dias and R. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals (to appear).
 [DSV]
K. Dalrymple, R. Strichartz and J. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5 (1999), 203284. MR 1683211 (2000k:31016)
 [DOS]
X. T. Duong, E. M. Ouhabaz and A. Sikora, Plancherel type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), 443485. MR 1943098 (2003k:43012)
 [FS]
M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal. 1 (1992), 135. MR 1245223 (95b:31009)
 [GHL]
A. Grigor'yan, Jiaxin Hu and KaSing Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 255 (2003), 20652095. MR 1953538 (2003j:60103)
 [HK]
B. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite selfsimilar fractals, Proc. London Math. Soc. 78 (3) (1999), 431458. MR 1665249 (99m:60118)
 [KZ]
S. Kasuoka and X. Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields 93 (1992), 169196. MR 1176724 (94e:60069)
 [Ki1]
J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 8 (1989), 259290. MR 1001286 (91g:31005)
 [K2]
J. Kigami, Harmonic calculus on p.c.f. selfsimilar sets, Trans. Amer. Math. Soc. 335 (1993), 721755. MR 1076617 (93d:39008)
 [Ki3]
J. Kigami, Analysis on Fractals, Cambridge University Press, New York, 2001. MR 1840042 (2002c:28015)
 [Ki4]
J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), 399444. MR 2017320
 [M]
V. Metz, Renormalization contracts on nested fractals, J. Reine Angew. Math. 480 (1996), 161175. MR 1420562 (98b:31007)
 [OSS]
R. Oberlin, B. Street and R. Strichartz, Sampling on the Sierpinski gasket, Experimental Math. 12 (2003), 403418.
 [P]
R. Peirone, Convergence and uniqueness problems for Dirichlet forms on fractals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), 431460. MR 1769995 (2001i:31016)
 [Sa]
C. Sabot, Existence and uniqueness of diffusions on finitely ramified selfsimilar fractals, Ann. Sci. Écoel Norm. Sup (4) 30 (1997), 605673. MR 1474807 (98h:60118)
 [SST]
J. Stanley, R. Strichartz and A. Teplayev, Energy partition on fractals, Indiana Univ. Math. Journal 52 (2003), 133156. MR 1970024 (2004a:31006)
 [SB]
L. N. Slobodeckii and V. M. Babic, On bounds for the Dirichlet integrals (Russian), Doklady Akad. Nauk. S.S.S.R. 106 (1956), 604606. MR 0076886 (17:959d)
 [S1]
R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 10311060. MR 0215084 (35:5927)
 [S2]
R. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal. 164 (1999), 181208. MR 1695571 (2000f:35032)
 [S3]
R. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46 (1999), 11991208. MR 1715511 (2000i:58035)
 [S4]
R. Strichartz, The Laplacian on the Sierpinski gasket via the method of averages, Pac. J. Math. 201 (2001), 241256. MR 1867899 (2003f:35056)
 [S5]
R. Strichartz, Function spaces on fractals, J. Funct. Anal. 198 (2003), 4383.MR 1962353 (2003m:46058)
 [S6]
R. Strichartz, Fractafolds based on the Sierpinski gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 40194043. MR 1990573 (2004b:28013)
 [SU]
R. Strichartz and M. Usher, Splines on fractals, Math. Proc. Cambridge Phil. Soc. 129 (2000), 331. MR 1765920 (2001c:28016)
 [B]
 G. Barbatis, Explicit estimates on the fundamental solution of higherorder parabolic equations with measurable coefficients, J. Diff. Eq. 174 (2001), 442463. MR 1846743 (2002i:35002)
 [BD]
 G. Barbatis and E. B. Davies, Sharp bounds on heat kernels of higher order uniform operators, J. Operator Theory 36 (1996), 179198.MR 1417193 (97k:35105)
 [Ba]
 M. Barlow, Diffusion on fractals, Lecture Notes Math., vol. 1690, Springer, 1998.MR 1668115 (2000a:60148)
 [BP]
 M. Barlow and E. Perkins, Brownian motion on the Sierpinski gasket, Probab. Theory Related Fields 79 (1988), 543623. MR 0966175 (89g:60241)
 [BST]
 O. BenBassat, R. Strichartz and A. Teplyaev, What is not in the domain of the Laplacian on Sierpinski gasket type fractals, J. of Functional Analysis 166 (1999), 197217. MR 1707752 (2001e:31016)
 [BSSY]
 N. BenGal, A. ShawKrauss, R. Strichartz and C. Young, Calculus on the Sierpinski gasket II: point singularities, eigenfunctions, and normal derivatives of the heat kernel, preprint.
 [Be]
 C. Berg, Potential theory on the infinite dimensional torus, Inv. Math. 32 (1976), 49100. MR 0402093 (53:5915)
 [BrP]
 J. H. Bramble and L.E. Payne, Bounds for the first derivatives of Green's function, Atti Accad. Naz. Lincei. Rend. Cl. Sci. Fis. Mat. Natur (8) 42 (1967), 604610. MR 0224850 (37:449)
 [C]
 T. Coulhon, Offdiagonal heat kernel lower bounds without Poincaré, J. London Math. Soc. (2) 68 (2003), 795816. MR 2010012
 [CDS]
 K. Coletta, K. Dias and R. Strichartz, Numerical analysis on the Sierpinski gasket, with applications to Schrödinger equations, wave equation, and Gibbs' phenomenon, Fractals (to appear).
 [DSV]
 K. Dalrymple, R. Strichartz and J. Vinson, Fractal differential equations on the Sierpinski gasket, J. Fourier Anal. Appl. 5 (1999), 203284. MR 1683211 (2000k:31016)
 [DOS]
 X. T. Duong, E. M. Ouhabaz and A. Sikora, Plancherel type estimates and sharp spectral multipliers, J. Funct. Anal. 196 (2002), 443485. MR 1943098 (2003k:43012)
 [FS]
 M. Fukushima and T. Shima, On a spectral analysis for the Sierpinski gasket, Potential Anal. 1 (1992), 135. MR 1245223 (95b:31009)
 [GHL]
 A. Grigor'yan, Jiaxin Hu and KaSing Lau, Heat kernels on metric measure spaces and an application to semilinear elliptic equations, Trans. Amer. Math. Soc. 255 (2003), 20652095. MR 1953538 (2003j:60103)
 [HK]
 B. Hambly and T. Kumagai, Transition density estimates for diffusion processes on post critically finite selfsimilar fractals, Proc. London Math. Soc. 78 (3) (1999), 431458. MR 1665249 (99m:60118)
 [KZ]
 S. Kasuoka and X. Y. Zhou, Dirichlet forms on fractals: Poincaré constant and resistance, Probab. Theory Related Fields 93 (1992), 169196. MR 1176724 (94e:60069)
 [Ki1]
 J. Kigami, A harmonic calculus on the Sierpinski spaces, Japan J. Appl. Math. 8 (1989), 259290. MR 1001286 (91g:31005)
 [K2]
 J. Kigami, Harmonic calculus on p.c.f. selfsimilar sets, Trans. Amer. Math. Soc. 335 (1993), 721755. MR 1076617 (93d:39008)
 [Ki3]
 J. Kigami, Analysis on Fractals, Cambridge University Press, New York, 2001. MR 1840042 (2002c:28015)
 [Ki4]
 J. Kigami, Harmonic analysis for resistance forms, J. Funct. Anal. 204 (2003), 399444. MR 2017320
 [M]
 V. Metz, Renormalization contracts on nested fractals, J. Reine Angew. Math. 480 (1996), 161175. MR 1420562 (98b:31007)
 [OSS]
 R. Oberlin, B. Street and R. Strichartz, Sampling on the Sierpinski gasket, Experimental Math. 12 (2003), 403418.
 [P]
 R. Peirone, Convergence and uniqueness problems for Dirichlet forms on fractals, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8) 3 (2000), 431460. MR 1769995 (2001i:31016)
 [Sa]
 C. Sabot, Existence and uniqueness of diffusions on finitely ramified selfsimilar fractals, Ann. Sci. Écoel Norm. Sup (4) 30 (1997), 605673. MR 1474807 (98h:60118)
 [SST]
 J. Stanley, R. Strichartz and A. Teplayev, Energy partition on fractals, Indiana Univ. Math. Journal 52 (2003), 133156. MR 1970024 (2004a:31006)
 [SB]
 L. N. Slobodeckii and V. M. Babic, On bounds for the Dirichlet integrals (Russian), Doklady Akad. Nauk. S.S.S.R. 106 (1956), 604606. MR 0076886 (17:959d)
 [S1]
 R. Strichartz, Multipliers on fractional Sobolev spaces, J. Math. Mech. 16 (1967), 10311060. MR 0215084 (35:5927)
 [S2]
 R. Strichartz, Some properties of Laplacians on fractals, J. Funct. Anal. 164 (1999), 181208. MR 1695571 (2000f:35032)
 [S3]
 R. Strichartz, Analysis on fractals, Notices Amer. Math. Soc. 46 (1999), 11991208. MR 1715511 (2000i:58035)
 [S4]
 R. Strichartz, The Laplacian on the Sierpinski gasket via the method of averages, Pac. J. Math. 201 (2001), 241256. MR 1867899 (2003f:35056)
 [S5]
 R. Strichartz, Function spaces on fractals, J. Funct. Anal. 198 (2003), 4383.MR 1962353 (2003m:46058)
 [S6]
 R. Strichartz, Fractafolds based on the Sierpinski gasket and their spectra, Trans. Amer. Math. Soc. 355 (2003), 40194043. MR 1990573 (2004b:28013)
 [SU]
 R. Strichartz and M. Usher, Splines on fractals, Math. Proc. Cambridge Phil. Soc. 129 (2000), 331. MR 1765920 (2001c:28016)
Similar Articles
Retrieve articles in Transactions of the American Mathematical Society
with MSC (2000):
31C45,
28A80
Retrieve articles in all journals
with MSC (2000):
31C45,
28A80
Additional Information
Robert S. Strichartz
Affiliation:
Department of Mathematics, Malott Hall, Cornell University, Ithaca, New York 14853
Email:
str@math.cornell.edu
DOI:
http://dx.doi.org/10.1090/S0002994704036852
PII:
S 00029947(04)036852
Received by editor(s):
July 8, 2003
Published electronically:
September 23, 2004
Additional Notes:
The author’s research was supported in part by the National Science Foundation, grant DMS–0140194
Article copyright:
© Copyright 2004 American Mathematical Society
