Hokkaido Mathematical Journal

Discrete Green Potentials with Finite Energy

Hisayasu KURATA and Maretsugu YAMASAKI

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For a hyperbolic infinite network, it is well-known that Green potentials with finite energy are Dirichlet potentials. Conversely, if a Dirichlet potential has non-positive Laplacian, then it is a Green potential with finite energy. In this paper, we study whether a Dirichlet potential can be expressed as a difference of two Green potentials with finite energy. Comparisons of the Dirichlet sum of a function and that of its Laplacian play important roles in our study. As a by-product, we obtain a Riesz decomposition of a function whose Laplacian is a Dirichlet function.

Article information

Hokkaido Math. J., Volume 47, Number 3 (2018), 607-624.

First available in Project Euclid: 26 September 2018

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 31C20: Discrete potential theory and numerical methods
Secondary: 31C25: Dirichlet spaces

discrete potential theory Dirichlet potential Green potential Riesz representation discrete Laplacian


KURATA, Hisayasu; YAMASAKI, Maretsugu. Discrete Green Potentials with Finite Energy. Hokkaido Math. J. 47 (2018), no. 3, 607--624. doi:10.14492/hokmj/1537948833. https://projecteuclid.org/euclid.hokmj/1537948833

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