Symmetric Harmonic Groups and Translation Invariant Dirichlet Spaces.
Gunnar Forst (1972)
Inventiones mathematicae
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Gunnar Forst (1972)
Inventiones mathematicae
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It is proved that the space of solutions of the Dirichlet problem for the harmonic functions in the unit disk with nontangential boundary limits 0 a.e. has the infinite dimension.
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We consider post-critically finite self-similar fractals with regular harmonic structures. We first obtain effective resistance estimates in terms of the Euclidean metric, which in particular imply the embedding theorem for the domains of the Dirichlet forms associated with the harmonic structures. We then characterize the domains of the Dirichlet forms.
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