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1. CJM 2005 (vol 57 pp. 598)

Kornelson, Keri A.
 Local Solvability of Laplacian Difference Operators Arising from the Discrete Heisenberg Group 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 operatorCategories:43A85, 22D10, 39A70

2. CJM 2003 (vol 55 pp. 401)

Varopoulos, N. Th.
 Gaussian Estimates in Lipschitz Domains We give upper and lower Gaussian estimates for the diffusion kernel of a divergence and nondivergence form elliptic operator in a Lipschitz domain. On donne des estimations Gaussiennes pour le noyau d'une diffusion, r\'eversible ou pas, dans un domaine Lipschitzien. Categories:39A70, 35-02, 65M06

3. CJM 2001 (vol 53 pp. 1057)

Varopoulos, N. Th.
 Potential Theory in Lipschitz Domains We prove comparison theorems for the probability of life in a Lipschitz domain between Brownian motion and random walks. On donne des th\'eor\`emes de comparaison pour la probabilit\'e de vie dans un domain Lipschitzien entre le Brownien et de marches al\'eatoires. Categories:39A70, 35-02, 65M06

4. CJM 1997 (vol 49 pp. 944)

Jia, R. Q.; Riemenschneider, S. D.; Zhou, D. X.
 Approximation by multiple refinable functions We consider the shift-invariant space, $\bbbs(\Phi)$, generated by a set $\Phi=\{\phi_1,\ldots,\phi_r\}$ of compactly supported distributions on $\RR$ when the vector of distributions $\phi:=(\phi_1,\ldots,\phi_r)^T$ satisfies a system of refinement equations expressed in matrix form as $$\phi=\sum_{\alpha\in\ZZ}a(\alpha)\phi(2\,\cdot - \,\alpha)$$ where $a$ is a finitely supported sequence of $r\times r$ matrices of complex numbers. Such {\it multiple refinable functions} occur naturally in the study of multiple wavelets. The purpose of the present paper is to characterize the {\it accuracy} of $\Phi$, the order of the polynomial space contained in $\bbbs(\Phi)$, strictly in terms of the refinement mask $a$. The accuracy determines the $L_p$-approximation order of $\bbbs(\Phi)$ when the functions in $\Phi$ belong to $L_p(\RR)$ (see Jia~[10]). The characterization is achieved in terms of the eigenvalues and eigenvectors of the subdivision operator associated with the mask $a$. In particular, they extend and improve the results of Heil, Strang and Strela~[7], and of Plonka~[16]. In addition, a counterexample is given to the statement of Strang and Strela~[20] that the eigenvalues of the subdivision operator determine the accuracy. The results do not require the linear independence of the shifts of $\phi$. Keywords:Refinement equations, refinable functions, approximation, order, accuracy, shift-invariant spaces, subdivisionCategories:39B12, 41A25, 65F15
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