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

[摘要] Differential operators $D_x$, $D_y$, and $D_z$ are formed using theaction of the $3$-dimensional discrete Heisenberg group $G$ on a set$S$, and the operators will act on functions on $S$. The Laplacianoperator $L=D_x^2 + D_y^2 + D_z^2$ is a difference operator withvariable differences which can be associated to a unitaryrepresentation of $G$ on the Hilbert space $L^2(S)$. Using techniquesfrom harmonic analysis and representation theory, we show that theLaplacian operator is locally solvable.

[学科分类] 数学（综合）