# Partial differential operators depending analytically on a parameter

• Volume: 41, Issue: 3, page 577-599
• ISSN: 0373-0956

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## Abstract

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Let $P\left(\lambda ,D\right)={\sum }_{|\alpha |\le m}{a}_{\alpha }\left(\lambda \right){D}^{\alpha }$ be a differential operator with constant coefficients ${a}_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\left\{$ P($\lambda$,D)$\right\}$ is of constant strength. We investigate the equation $P\left(\lambda ,D\right){𝔣}_{\lambda }\equiv {g}_{\lambda }$ where ${𝔤}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${𝔣}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander.

## How to cite

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Mantlik, Frank. "Partial differential operators depending analytically on a parameter." Annales de l'institut Fourier 41.3 (1991): 577-599. <http://eudml.org/doc/74930>.

@article{Mantlik1991,
abstract = {Let $P(\lambda ,D)=\sum _\{\vert \alpha \vert \le m\}a_\{\alpha \}(\lambda )D^\{\alpha \}$ be a differential operator with constant coefficients $a_\{\alpha \}$ depending analytically on a parameter $\lambda$. Assume that the family $\lbrace$ P($\lambda$,D)$\rbrace$ is of constant strength. We investigate the equation $P(\lambda ,D)\{\frak f\}_\{\lambda \}\equiv g_\{\lambda \}$ where $\{\frak g\}_\{\lambda \}$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution $\{\frak f\}_\{\lambda \}$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander.},
author = {Mantlik, Frank},
journal = {Annales de l'institut Fourier},
keywords = {linear differential operator; analytic dependence; elementary solutions; constant coefficients; constant strength; regular fundamental solution},
language = {eng},
number = {3},
pages = {577-599},
publisher = {Association des Annales de l'Institut Fourier},
title = {Partial differential operators depending analytically on a parameter},
url = {http://eudml.org/doc/74930},
volume = {41},
year = {1991},
}

TY - JOUR
AU - Mantlik, Frank
TI - Partial differential operators depending analytically on a parameter
JO - Annales de l'institut Fourier
PY - 1991
PB - Association des Annales de l'Institut Fourier
VL - 41
IS - 3
SP - 577
EP - 599
AB - Let $P(\lambda ,D)=\sum _{\vert \alpha \vert \le m}a_{\alpha }(\lambda )D^{\alpha }$ be a differential operator with constant coefficients $a_{\alpha }$ depending analytically on a parameter $\lambda$. Assume that the family $\lbrace$ P($\lambda$,D)$\rbrace$ is of constant strength. We investigate the equation $P(\lambda ,D){\frak f}_{\lambda }\equiv g_{\lambda }$ where ${\frak g}_{\lambda }$ is a given analytic function of $\lambda$ with values in some space of distributions and the solution ${\frak f}_{\lambda }$ is required to depend analytically on $\lambda$, too. As a special case we obtain a regular fundamental solution of P($\lambda$,D) which depends analytically on $\lambda$. This result answers a question of L. Hörmander.
LA - eng
KW - linear differential operator; analytic dependence; elementary solutions; constant coefficients; constant strength; regular fundamental solution
UR - http://eudml.org/doc/74930
ER -

## References

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1. [G] H. GRAUERT, On Levi's problem and the embedding of real-analytic manifolds, Ann. Math., 68, 2 (1958), 460-472. Zbl0108.07804MR20 #5299
2. [H1] L. HÖRMANDER, The analysis of linear partial differential operators I, Grundlehren d. mathem. Wissensch., 256, Springer (1983). Zbl0521.35001MR85g:35002a
3. [II2] L. HÖRMANDER, The analysis of linear partial differential operators II, Grundlehren d. mathem. Wissensch., 257, Springer (1983). Zbl0521.35002MR85g:35002b
4. [L] J. LEITERER, Banach coherent analytic Fréchet sheaves, Math. Nachr., 85 (1978), 91-109. Zbl0409.32017MR80b:32026
5. [M] F. MANTLIK, Fundamental solutions for hypoelliptic differential operators depending analytically on a parameter, to appear. Zbl0785.35008
6. [T1] F. TRÈVES, Un théorème sur les équations aux dérivées partielles à coefficients constants dépendant de paramètres, Bull. Soc. Math. France, 90 (1962), 473-486. Zbl0113.30901MR26 #6582
7. [T2] F. TRÈVES, Fundamental solutions of linear partial differential equations with constant coefficients depending on parameters, Am. J. Math., 84 (1962), 561-577. Zbl0121.32201MR26 #6580

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