The coherence of complemented ideals in the space of real analytic functions
If \( V \) is a complex analytic subvariety of a complex neighborhood of \( \mathbb R^d \), then its ideal \( J_V(\mathbb R^d) \) is defined as the set of all real analytic functions on \( \mathbb R^d \) such that any extension to a holomorphic function on a neighborhood of \( \mathbb R^d \) vanishes in a neighborhood, relative to \( V \), of \( X = V \cap \mathbb R^d \). It is shown that if \( J_V(\mathbb R^d) \) is linear topologically complemented in the space \( \mathcal A(\mathbb R^d) \) of all real analytic functions on \( \mathbb R^d \), then \( V \) is the smallest complex analytic variety among those which have the same real part and, moreover, \(V\) satisfies the condition \(\text{PL}_{\text{loc}}\) at every real singular point. If \(V\) is homogeneous or \(X\) is compact, then the converse also holds. The local Phragmén-Lindelöf condition \(\text{PL}_{\text{loc}}\) was introduced by \textit{L.\,Hörmander} in the investigation of solvability problems for partial differential equations [Invent.\ Math.\ 21, 151--182 (1973; Zbl 0282.35015)]. The paper extends earlier results of \textit{D.\,Vogt}, in particular [J.~Reine Angew.\ Math.\ 606, 217--233 (2007; Zbl 1133.46014)]. In the present paper, a change of perspective with regard to the real analytic variety \( X \) leads to the surprising result that the coherence requirements in the older paper are automatically satisfied for complemented ideals.
scientific article; zbMATH DE number 5700558
scientific article