# Localizations of partial differential operators and surjectivity on real analytic functions

Studia Mathematica (2000)

• Volume: 140, Issue: 1, page 15-40
• ISSN: 0039-3223

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## Abstract

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Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $\Omega \subset {ℝ}^{n}$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization ${P}_{m,\Theta }$ of the principal part ${P}_{m}$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for ${P}_{m,\Theta }$. Under additional assumptions ${P}_{m}$ must be locally hyperbolic.

## How to cite

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Langenbruch, Michael. "Localizations of partial differential operators and surjectivity on real analytic functions." Studia Mathematica 140.1 (2000): 15-40. <http://eudml.org/doc/216753>.

@article{Langenbruch2000,
abstract = {Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $Ω ⊂ ℝ^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_\{m,Θ\}$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_\{m,Θ\}$. Under additional assumptions $P_m$ must be locally hyperbolic.},
author = {Langenbruch, Michael},
journal = {Studia Mathematica},
keywords = {partial differential operator; real analytic function; elementary solution; hyperbolicity; local hyperbolicity; analytic surjectivity},
language = {eng},
number = {1},
pages = {15-40},
title = {Localizations of partial differential operators and surjectivity on real analytic functions},
url = {http://eudml.org/doc/216753},
volume = {140},
year = {2000},
}

TY - JOUR
AU - Langenbruch, Michael
TI - Localizations of partial differential operators and surjectivity on real analytic functions
JO - Studia Mathematica
PY - 2000
VL - 140
IS - 1
SP - 15
EP - 40
AB - Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $Ω ⊂ ℝ^n$. Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m,Θ}$ of the principal part $P_m$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m,Θ}$. Under additional assumptions $P_m$ must be locally hyperbolic.
LA - eng
KW - partial differential operator; real analytic function; elementary solution; hyperbolicity; local hyperbolicity; analytic surjectivity
UR - http://eudml.org/doc/216753
ER -

## References

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