Domains of Dirichlet forms and effective resistance estimates on p.c.f. fractals

Jiaxin Hu; Xingsheng Wang

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Infinite dimension of solutions of the Dirichlet problem

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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.

Dirichlet problem with L $^{p}$ -boundary data in contractible domains of Carnot groups

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Let $ℒ$ be a sub-laplacian on a stratified Lie group $G$ . In this paper we study the Dirichlet problem for $ℒ$ with $L^{p}$ -boundary data, on domains $Ω$ which are contractible with respect to the natural dilations of $G$ . One of the main difficulties we face is the presence of non-regular boundary points for the usual Dirichlet problem for $ℒ$ . A potential theory approach is followed. The main results are applied to study a suitable notion of Hardy spaces.

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Growth of coefficients of universal Dirichlet series

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