On a closed range property of a linear differential operator (Q1085335)

The following fact is shown: For a compact set \(K\subset {\mathbb{R}}^ n\) and a linear differential operator \(P(x,D_ x)\) defined on a neighborhood of K, P\({\mathcal A}(K)\) is a closed subspace of \({\mathcal A}(K)\) under some geometric condition on the pair (K,P), which the authors call the uniform P-convexity of K. Here \({\mathcal A}(K)\) denotes the space of real analytic functions on K. The reasoning is based on a result on the finite-dimensionality of some cohomology groups associated with an elliptic system of linear differential equations [see \textit{T. Kawai}, ibid. 49, 243-246 (1973; Zbl 0303.35056)]. The elliptic system used in this article is determined by the operator \(P(x,D_ x)\) and the Cauchy-Riemann operators. One interesting fact is that the notion of uniform P-convexity is quite akin to that of the strong P-convexity which \textit{L. Hörmander} [''Linear Partial Differential Operators'' (1963; Zbl 0108.093)] used to obtain a priori estimates of solutions. A similar result had been claimed by \textit{S. Kiro} [On the global existence of holomorphic solutions and the semiglobal existence of real analytic solutions of linear partial differential equations, Weizmann Institute Preprint Rehovot, Israel], and it is used in the first-named author's article titled ''On the global existence of real analytic solutions and hyperfunction solutions of linear differential equations, Proc. Japan Acad., Ser. A 62, 77-79 (1986). However, Kiro's reasoning is erroneous, and hence the condition (1.2) in the above article of Kawai should be replaced by (1) in this reviewed article.