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Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
Published online by Cambridge University Press: 20 November 2018
Abstract
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Differential operators 
${{D}_{x,}}\,{{D}_{y}}$, and 
${{D}_{z}}$ are formed using the action of the 3-dimensional discrete Heisenberg group 
$G$ on a set 
$S$, and the operators will act on functions on $S$. The Laplacian operator 
$L\,=\,D_{x}^{2}+D_{y}^{2}+D_{z}^{2}$ is a difference operator with variable differences which can be associated to a unitary representation of $G$ on the Hilbert space 
${{L}^{2}}\left( S \right)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.
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- Copyright © Canadian Mathematical Society 2005
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