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

Michael Langenbruch. "Characterization of surjective partial differential operators on spaces of real analytic functions." Studia Mathematica 162.1 (2004): 53-96. <http://eudml.org/doc/284717>.

@article{MichaelLangenbruch2004,
abstract = {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.},
author = {Michael Langenbruch},
journal = {Studia Mathematica},
keywords = {derived functors; DFS-spaces; elementary solution; analytic singular support},
language = {eng},
number = {1},
pages = {53-96},
title = {Characterization of surjective partial differential operators on spaces of real analytic functions},
url = {http://eudml.org/doc/284717},
volume = {162},
year = {2004},
}

TY - JOUR
AU - Michael Langenbruch
TI - Characterization of surjective partial differential operators on spaces of real analytic functions
JO - Studia Mathematica
PY - 2004
VL - 162
IS - 1
SP - 53
EP - 96
AB - 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.
LA - eng
KW - derived functors; DFS-spaces; elementary solution; analytic singular support
UR - http://eudml.org/doc/284717
ER -