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A harmonic calculus on the Sierpiński spaces. (English) Zbl 0686.31003
Starting with an equilateral triangle in \(R^ 2\) and successively removing, ad infinitum, equilateral triangles whose vertices are midpoints of the equilateral triangles of the preceding generation, W. Sierpiński obtained a gasket-like compact metric space, here denoted \(K^ 3\subset R^ 2\). The present author generalizes this construction starting with a regular (equilateral) simplex in \(R^{N-1}\) to obtain \(K^ N\subset R^{N-1}\) of Hausdorff dimension \(\log N/\log 2.\) This paper is devoted to the construction of a theory of harmonic functions on \(K^ N\), called harmonic calculus. The topics covered are harmonic differences, harmonic functions and their series expansion. The latter notions are used to study the analogues of Laplace operator, Poisson equation, Dirichlet problem, Neumann derivatives and Gauss-Green formula. The exposition is very painstaking. It is surprising to this reviewer that such results similar to the classical, smooth case are valid in the present context.
Reviewer: E.J.Akutowicz

MSC:
31B20 Boundary value and inverse problems for harmonic functions in higher dimensions
35J05 Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation
55M10 Dimension theory in algebraic topology
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