Let A(Ω) denote the real analytic functions defined on an open set Ω ⊂ ℝⁿ. We show that a partial differential operator P(D) with constant coefficients is surjective on A(Ω) if and only if for any relatively compact open ω ⊂ Ω, P(D) admits (shifted) hyperfunction elementary solutions on Ω which are real analytic on ω (and if the equation P(D)f = g, g ∈ A(Ω), may be solved on ω). The latter condition is redundant if the elementary solutions are defined on conv(Ω). This extends and improves previous results of Andersson, Kawai, Kaneko and Zampieri. For convex Ω, a different characterization of surjective operators P(D) on A(Ω) was given by Hörmander using a Phragmén-Lindelöf type condition, which cannot be extended to the case of noncovex Ω. 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-1-4,
author = {Michael Langenbruch},
title = {Characterization of surjective partial differential operators on spaces of real analytic functions},
journal = {Studia Mathematica},
volume = {162},
year = {2004},
pages = {53-96},
zbl = {1073.35054},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-1-4}
}
Michael Langenbruch. Characterization of surjective partial differential operators on spaces of real analytic functions. Studia Mathematica, Tome 162 (2004) pp. 53-96. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm162-1-4/