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On the local solvability of vector fields with critical points
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 10 / Issue 3 / July 2011
- Published online by Cambridge University Press:
- 12 May 2011, pp. 785-808
- Print publication:
- July 2011
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Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group
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- Journal:
- Canadian Journal of Mathematics / Volume 57 / Issue 3 / 01 June 2005
- Published online by Cambridge University Press:
- 20 November 2018, pp. 598-615
- Print publication:
- 01 June 2005
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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.
