Division et extension dans des classes de Carleman de fonctions holomorphes

Vincent Thilliez

Banach Center Publications (1998)

  • Volume: 44, Issue: 1, page 233-246
  • ISSN: 0137-6934

Abstract

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Let Ω be a bounded pseudoconvex domain in ℂ n with C 1 boundary and let X be a complete intersection submanifold of Ω, defined by holomorphic functions v 1 , . . . , v p (1 ≤ p ≤ n-1) smooth up to ∂Ω. We give sufficient conditions ensuring that a function f holomorphic in X (resp. in Ω, vanishing on X), and smooth up to the boundary, extends to a function g holomorphic in Ω and belonging to a given strongly non-quasianalytic Carleman class l ! M l in Ω ¯ (resp. satisfies f = v 1 f 1 + . . . + v p f p with f 1 , . . . , f p holomorphic in Ω and l ! M l -regular in Ω ¯ ). The essential assumption is that f and v 1 , . . . , v p belong to some (maybe smaller) Carleman class l ! M l - , where the sequences M - and M are precisely related by geometric conditions on X and Ω.

How to cite

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Thilliez, Vincent. "Division et extension dans des classes de Carleman de fonctions holomorphes." Banach Center Publications 44.1 (1998): 233-246. <http://eudml.org/doc/208887>.

@article{Thilliez1998,
abstract = {Let Ω be a bounded pseudoconvex domain in $ℂ^n$ with $C^1$ boundary and let X be a complete intersection submanifold of Ω, defined by holomorphic functions $v_1,...,v_p$ (1 ≤ p ≤ n-1) smooth up to ∂Ω. We give sufficient conditions ensuring that a function f holomorphic in X (resp. in Ω, vanishing on X), and smooth up to the boundary, extends to a function g holomorphic in Ω and belonging to a given strongly non-quasianalytic Carleman class $\{l!M_l\}$ in $\bar\{Ω\}$ (resp. satisfies $f = v_1 f_1+... + v_p f_p$ with $f_1,...,f_p$ holomorphic in Ω and $\{l!M_l\}$-regular in $\bar\{Ω\}$). The essential assumption is that f and $v_1,... ,v_p$ belong to some (maybe smaller) Carleman class $\{l!M^-_l\}$, where the sequences $M^-$ and M are precisely related by geometric conditions on X and Ω.},
author = {Thilliez, Vincent},
journal = {Banach Center Publications},
keywords = {extension; Carleman class of holomorphic functions; division theorem},
language = {eng},
number = {1},
pages = {233-246},
title = {Division et extension dans des classes de Carleman de fonctions holomorphes},
url = {http://eudml.org/doc/208887},
volume = {44},
year = {1998},
}

TY - JOUR
AU - Thilliez, Vincent
TI - Division et extension dans des classes de Carleman de fonctions holomorphes
JO - Banach Center Publications
PY - 1998
VL - 44
IS - 1
SP - 233
EP - 246
AB - Let Ω be a bounded pseudoconvex domain in $ℂ^n$ with $C^1$ boundary and let X be a complete intersection submanifold of Ω, defined by holomorphic functions $v_1,...,v_p$ (1 ≤ p ≤ n-1) smooth up to ∂Ω. We give sufficient conditions ensuring that a function f holomorphic in X (resp. in Ω, vanishing on X), and smooth up to the boundary, extends to a function g holomorphic in Ω and belonging to a given strongly non-quasianalytic Carleman class ${l!M_l}$ in $\bar{Ω}$ (resp. satisfies $f = v_1 f_1+... + v_p f_p$ with $f_1,...,f_p$ holomorphic in Ω and ${l!M_l}$-regular in $\bar{Ω}$). The essential assumption is that f and $v_1,... ,v_p$ belong to some (maybe smaller) Carleman class ${l!M^-_l}$, where the sequences $M^-$ and M are precisely related by geometric conditions on X and Ω.
LA - eng
KW - extension; Carleman class of holomorphic functions; division theorem
UR - http://eudml.org/doc/208887
ER -

References

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  13. [Th1] V. Thilliez, Prolongement dans des classes ultradifférentiables et propriétés de régularité des compacts de ℝ n , Ann. Polon. Math. 63 (1996), 71-88. 
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