Division et extension dans des classes de Carleman de fonctions holomorphes

Let Ω be a bounded pseudoconvex domain in ${\displaystyle {ℂ}^n}$ with ${\displaystyle C^1}$ boundary and let X be a complete intersection submanifold of Ω, defined by holomorphic functions ${\displaystyle v_1,...,v_p}$ (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 ${\displaystyle \{l!M_l\}}$ in ${\displaystyle \overline{\Omega }}$ (resp. satisfies ${\displaystyle f=v_1{f}_1+...+v_p{f}_p}$ with ${\displaystyle f_1,...,f_p}$ holomorphic in Ω and ${\displaystyle \{l!M_l\}}$-regular in ${\displaystyle \overline{\Omega }}$). The essential assumption is that f and ${\displaystyle v_1,...,v_p}$ belong to some (maybe smaller) Carleman class ${\displaystyle \{l!M^{-}\_l\}}$, where the sequences ${\displaystyle M^{-}}$ and M are precisely related by geometric conditions on X and Ω.