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Extension Gevrey et rigidité dans un secteur

Vincent Thilliez — 1995

Studia Mathematica

We study a rigidity property, at the vertex of some plane sector, for Gevrey classes of holomorphic functions in the sector. For this purpose, we prove a linear continuous version of Borel-Ritt's theorem with Gevrey conditions

Division et extension dans des classes de Carleman de fonctions holomorphes

Vincent Thilliez — 1998

Banach Center Publications

Let Ω be a bounded pseudoconvex domain in 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...

Bounds for quotients in rings of formal power series with growth constraints

Vincent Thilliez — 2002

Studia Mathematica

In rings Γ M of formal power series in several variables whose growth of coefficients is controlled by a suitable sequence M = ( M l ) l 0 (such as rings of Gevrey series), we find precise estimates for quotients F/Φ, where F and Φ are series in Γ M such that F is divisible by Φ in the usual ring of all power series. We give first a simple proof of the fact that F/Φ belongs also to Γ M , provided Γ M is stable under derivation. By a further development of the method, we obtain the main result of the paper, stating that...

Łojasiewicz ideals in Denjoy-Carleman classes

Vincent Thilliez — 2013

Studia Mathematica

The classical notion of Łojasiewicz ideals of smooth functions is studied in the context of non-quasianalytic Denjoy-Carleman classes. In the case of principal ideals, we obtain a characterization of Łojasiewicz ideals in terms of properties of a generator. This characterization involves a certain type of estimates that differ from the usual Łojasiewicz inequality. We then show that basic properties of Łojasiewicz ideals in the case have a Denjoy-Carleman counterpart.

Hilbert-valued forms and barriers on weakly pseudoconvex domains.

Vincent Thilliez — 1998

Publicacions Matemàtiques

We introduce an alternative proof of the existence of certain C barrier maps, with polynomial explosion of the derivatives, on weakly pseudoconvex domains in C. Barriers of this sort have been constructed very recently by J. Michel and M.-C. Shaw, and have various applications. In our paper, the adaptation of Hörmander's L techniques to suitable vector-valued functions allows us to give a very simple approach of the problem and to improve some aspects of the result of Michel and Shaw, regarding...

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