Analytic extension of non quasi-analytic Whitney jets of Roumieu type
The classical Whitney theorem asserts that every \({\mathcal C}^\infty\)-Whitney jet on a closed subset \(F\) of \(\mathbb{R}^n\) has a \({\mathcal C}^\infty\)-extension on \(\mathbb{R}^n\) which is analytic on \(\mathbb{R}^n\backslash F\). This result has become the starting point to numerous generalizations and specifications. In this paper, the authors prove a version of Whitney's theorem for Whitney jets of Roumieu type. To be more specific, let \((M_r)_{r\in\mathbb{N}_0}\), where \(M_0=1\), be a logarithmically convex, non-quasianalytic sequence of positive numbers, let \(F\) be a closed proper subset of \(\mathbb{R}^n\) and let \({\mathcal E}_{\{M_r\}}(\mathbb{R}^n)\) denote the non-quasianalytic \((M_r)\)-class of Roumieu type on \(\mathbb{R}^n\), i.e., the set of all \({\mathcal C}^\infty\)-functions on \(\mathbb{R}^n\) such that for every compact subset \(K\) of \(\mathbb{R}^n\) there is a constant \(h>0\) such that \(\sup_{\alpha\in\mathbb{N}^n_0}{h^{|\alpha|}\over M_{|\alpha|}} |D^\alpha f|_K<\infty\). Then the main result of the paper reads as follows. For every \(f\in{\mathcal E}_{\{M_r\}}(\mathbb{R}^n)\) and every closed subset \(F\) of \(\mathbb{R}^n\), there is \(g\in{\mathcal E}_{\{M_r\}}(\mathbb{R}^n)\) such that \(D^\alpha f(x)= D^\alpha g(x)\) for every \(\alpha\in\mathbb{N}^n_0\) and \(x\in F\), and \(g\) is analytic on \(\mathbb{R}^n\backslash F\). If \(F\) is a compact subset of \(\mathbb{R}^n\), the above result was earlier proved by \textit{M. Valdivia} [Math. Jap. 44, No. 3, 415-434 (1996; Zbl 0874.46027)]. However, the proof of the non-compact case is by no means a direct evolution of the Valdivia original reasoning.
scientific article; zbMATH DE number 1029210