Characterization of surjective partial differential operators on spaces of real analytic functions

Michael Langenbruch¹

¹ Department of Mathematics University of Oldenburg D-26111 Oldenburg, Germany

Studia Mathematica, Tome 162 (2004) no. 1, pp. 53-96

Résumé

Let $A({\mit \Omega })$ denote the real analytic functions defined on an open set ${\mit \Omega } \subset {{\mathbb R}}^n$. We show that a partial differential operator $P(D)$ with constant coefficients is surjective on $A({\mit \Omega })$ if and only if for any relatively compact open $\omega \subset {\mit \Omega }$, $P(D)$ admits (shifted) hyperfunction elementary solutions on ${\mit \Omega }$ which are real analytic on $\omega $ (and if the equation $P(D)f = g$, $g\in A({\mit \Omega })$, may be solved on $\omega $). The latter condition is redundant if the elementary solutions are defined on $\mathop {\rm conv}\nolimits ({\mit \Omega })$. This extends and improves previous results of Andersson, Kawai, Kaneko and Zampieri. For convex ${\mit \Omega }$, a different characterization of surjective operators $P(D)$ on $A({\mit \Omega })$ was given by Hörmander using a Phragmén–Lindelöf type condition, which cannot be extended to the case of noncovex ${\mit \Omega }$. The paper is based on a surjectivity criterion for exact sequences of projective (DFS)-spectra which improves earlier results of Braun and Vogt, and Frerick and Wengenroth.

Détail

DOI : 10.4064/sm162-1-4

Keywords: mit omega denote real analytic functions defined set mit omega subset mathbb partial differential operator constant coefficients surjective mit omega only relatively compact omega subset mit omega admits shifted hyperfunction elementary solutions mit omega which real analytic omega equation mit omega may solved omega latter condition redundant elementary solutions defined mathop conv nolimits mit omega extends improves previous results andersson kawai kaneko zampieri convex mit omega different characterization surjective operators mit omega given rmander using phragm lindel type condition which cannot extended noncovex mit omega paper based surjectivity criterion exact sequences projective dfs spectra which improves earlier results braun vogt frerick wengenroth