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

Published:2005-06-01
Printed: Jun 2005
• Keri A. Kornelson
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Abstract

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 $L^2(S)$. Using techniques from harmonic analysis and representation theory, we show that the Laplacian operator is locally solvable.
 Keywords: discrete Heisenberg group, unitary representation, local solvability, difference operator
 MSC Classifications: 43A85 - Analysis on homogeneous spaces 22D10 - Unitary representations of locally compact groups 39A70 - Difference operators [See also 47B39]