On solvability of linear difference equations in smooth and real analytic vector functions of several variables (Q1329559)

Consider the multidimensional equations \[ \sum^ q_{j = 1} A_ j (x)y(x + e_ j) = f(x),\;e_ j \in \mathbb{R}^ n \tag{*} \] where \(x \in \mathbb{R}^ n\) and \(A_ j : \mathbb{R}^ n \to \Hom (\mathbb{R}^ p, \mathbb{R}^ m)\), \(f:\mathbb{R}^ n \to \mathbb{R}^ m\) are given maps. The authors study the solvability of \((*)\) in the class \(C^ k\), \(1 \leq k \leq \omega\) under the condition that the coefficients \(A_ j(x)\) \((j = 1,2, \dots,q)\) and \(f(x)\) belong to \(C^ k\).