Analysis on products of fractals. (English) Zbl 1056.31006
On many finitely ramified fractals (p.c.f. fractals, to be precise) a fractal “Laplacian” is known to exist. Here products of such fractals and of their Laplacians are considered. Intuitively, the Laplacian of a single fractal is almost one dimensional and gives rise to ODEs, the Laplacian of a product is higher dimensional and gives rise to PDEs. The arguments mimic the construction of the square and its classical two dimensional Laplacian as the product of two lines and their classical one dimensional Laplacians.
The same strategy is used by N. Bouleau and F. Hirsch [Dirichlet forms and analysis on Wiener space. (de Gruyter Studies in Mathematics 14), Berlin: de Gruyter (1991; Zbl 0748.60046)] to construct tensor products of Dirichlet spaces. Once an energy (Dirichlet form) has been constructed, harmonic functions are defined to be energy minimizing. The transition kernel of the product is the product of the individual transition kernels like in [J. Bliedtner and W. Hansen, Potential theory (Springer) (1986; Zbl 0706.31001)]. Known heat kernel estimates on p.c.f. fractals from B. M. Hambly and T. Kumagai [Proc. Lond. Math. Soc., III. Ser. 78, No. 2, 431–458 (1999; Zbl 1027.60087)] are then used to estimate the product kernel.
Together with spectral representations and further assumptions this allows the author to construct trace and extension operators, a Poisson formula for harmonic functions and pointwise expressions for the energy and the Laplacian. An abundance of open problems and future directions is mentioned throughout the text.
The same strategy is used by N. Bouleau and F. Hirsch [Dirichlet forms and analysis on Wiener space. (de Gruyter Studies in Mathematics 14), Berlin: de Gruyter (1991; Zbl 0748.60046)] to construct tensor products of Dirichlet spaces. Once an energy (Dirichlet form) has been constructed, harmonic functions are defined to be energy minimizing. The transition kernel of the product is the product of the individual transition kernels like in [J. Bliedtner and W. Hansen, Potential theory (Springer) (1986; Zbl 0706.31001)]. Known heat kernel estimates on p.c.f. fractals from B. M. Hambly and T. Kumagai [Proc. Lond. Math. Soc., III. Ser. 78, No. 2, 431–458 (1999; Zbl 1027.60087)] are then used to estimate the product kernel.
Together with spectral representations and further assumptions this allows the author to construct trace and extension operators, a Poisson formula for harmonic functions and pointwise expressions for the energy and the Laplacian. An abundance of open problems and future directions is mentioned throughout the text.
Reviewer: Volker Metz (Bielefeld)
MSC:
| 31C05 | Harmonic, subharmonic, superharmonic functions on other spaces |
| 60J45 | Probabilistic potential theory |
| 31C25 | Dirichlet forms |
| 28A80 | Fractals |
Keywords:
fractal Laplacian; tensor products of Dirichlet spaces; transition kernel; heat kernel; product kernel; trace operators; extension operators; Poisson formula; harmonic functionsReferences:
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