Partial differential operators depending analytically on a parameter

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 -