A remark on formal solvability of partial differential equations with constant coefficients (Q1330267)

Summary: Let \(s\) be a formal series and \(p = p(D) = \sum_ \alpha a_ \alpha D^ \alpha\) a partial differential operator, with constant coefficients. We say that \(s\) is a formal solution of \(ps = 0\) if there exists \(f \in C^ \infty(\mathbb{R}^ n)\) such that \(\sum_ q [f^{(q)} (0)/q!] z^ q = s\) and \((p(D)f)^{(q)} (0) = 0\), \(\forall q \in\mathbb{N}^ n\) \((z = (z_ 1, \dots, z_ n))\). We give a characterization of formal solvability of \(p\) in terms of extension, like distributions, over all \(\mathbb{R}^ n\) of the elements of the space \(\{u \in D' (\mathbb{R}^ n \backslash \{0\})\): \(p(-D) u = 0\}\).