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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 $logN/log2·$ 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 (higher-dimensional potential theory) 35J05 Laplacian operator, reduced wave equation (Helmholtz equation), Poisson equation 55M10 Dimension theory (algebraic topology)