Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

• Volume: 40, Issue: 3, page 619-655
• ISSN: 0373-0956

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Abstract

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Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P\left(D\right)$ are characterized that admit a continuous linear right inverse on $ℰ\left(\Omega \right)$ or ${𝒟}^{\text{'}}\left(\Omega \right)$, $\Omega$ an open set in ${\mathbf{R}}^{n}$. For bounded $\Omega$ with ${C}^{1}$-boundary these properties are equivalent to $P\left(D\right)$ being very hyperbolic. For $\Omega ={\mathbf{R}}^{n}$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.

How to cite

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Taylor, B. A., Meise, R., and Vogt, Dietmar. "Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse." Annales de l'institut Fourier 40.3 (1990): 619-655. <http://eudml.org/doc/74890>.

@article{Taylor1990,
abstract = {Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P(D)$ are characterized that admit a continuous linear right inverse on $\{\cal E\}(\Omega )$ or $\{\cal D\}^\{\prime \}(\Omega )$, $\Omega$ an open set in $\{\bf R\}^ n$. For bounded $\Omega$ with $C^ 1$-boundary these properties are equivalent to $P(D)$ being very hyperbolic. For $\Omega =\{\bf R\}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.},
author = {Taylor, B. A., Meise, R., Vogt, Dietmar},
journal = {Annales de l'institut Fourier},
keywords = {constant coefficient partial differential operators; continuous linear right inverse; Phragmén-Lindelöf condition},
language = {eng},
number = {3},
pages = {619-655},
publisher = {Association des Annales de l'Institut Fourier},
title = {Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse},
url = {http://eudml.org/doc/74890},
volume = {40},
year = {1990},
}

TY - JOUR
AU - Taylor, B. A.
AU - Meise, R.
AU - Vogt, Dietmar
TI - Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse
JO - Annales de l'institut Fourier
PY - 1990
PB - Association des Annales de l'Institut Fourier
VL - 40
IS - 3
SP - 619
EP - 655
AB - Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P(D)$ are characterized that admit a continuous linear right inverse on ${\cal E}(\Omega )$ or ${\cal D}^{\prime }(\Omega )$, $\Omega$ an open set in ${\bf R}^ n$. For bounded $\Omega$ with $C^ 1$-boundary these properties are equivalent to $P(D)$ being very hyperbolic. For $\Omega ={\bf R}^n$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$.
LA - eng
KW - constant coefficient partial differential operators; continuous linear right inverse; Phragmén-Lindelöf condition
UR - http://eudml.org/doc/74890
ER -

References

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