Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group

Kornelson, Keri A.

Canadian journal of mathematics, Tome 57 (2005) no. 3, pp. 598-615

Résumé

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.

Détail

DOI : 10.4153/CJM-2005-024-7

Mots-clés : 43A85, 22D10, 39A70, discrete Heisenberg group, unitary representation, local solvability, difference operator

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