Whitney's extension theorem for ultradifferentiable functions of Beurling type
The authors introduce classes of non-quasianalytic functions \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\) similar to those treated by Beurling and Björck: Given a weight function \(\omega\) : \({\mathbb{R}}\to [0,\infty [\) (i.e. \(\omega\) is continuous, even, increasing on [0,\(\infty [\), satisfies \(\omega (0)=0\), lim \(\omega\) (t)\(=\infty\), \(\limsup_{t\to \infty}\omega (2t)/\omega (t)<\infty,\) \(\lim_{t\to \infty}\log t/\omega (t)=0,\int^{+\infty}_{-\infty}(\omega (t)/1+t^ 2)dt<\infty,\) and \(\phi\) : \(t\mapsto \omega (e^ t)\) is convex on \({\mathbb{R}})\), let \[ {\mathcal E}_{\omega}({\mathbb{R}}^ n):=\{f\in C^{\infty}({\mathbb{R}}^ n):\quad \forall_{m\in {\mathbb{N}}}\sup_{\alpha \in {\mathbb{N}}^ n_ 0}\sup_{| x| \leq m}| f^{(\alpha)}(x)| \exp (- m\phi^*(| \alpha | /m))<\infty \}, \] where \(\phi^*: x\mapsto \sup \{xy-\phi (y):\) \(y\geq 0\}\) denotes the Young conjugate of \(\phi\) \(| [0,\infty [.\) The main result of the present paper is the following version of Whitney's extension theorem - which includes also previous results of Ehrenpreis and Komatsu. Theorem (shortened form). For a weight function \(\omega\) the following are equivalent: (1) For all \(n\in {\mathbb{N}}\) and all \(K\subset {\mathbb{R}}^ n\) compact convex with \(\overset\circ K \neq \emptyset\) the restriction \(\rho_ K: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}(K)\), \(f\mapsto (f^{\alpha}| K)_{\alpha \in {\mathbb{N}}^ n_ 0}\) is surjective. (2) For all \(n\in {\mathbb{N}}\) and all bounded open sets \(\Omega \subset {\mathbb{R}}^ n\) with real-analytic boundary the map \(\rho_{{\bar \Omega}}: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}({\bar \Omega})\) is surjective. (3) For all \(n\in {\mathbb{N}}\) the Borel map \(B_ n: {\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}\{0\}\), \(B_ n(f):=(f^{(\alpha)}(0))_{\alpha \in {\mathbb{N}}^ n_ 0}\), is surjective. (4) There exists \(C>0\) with \(\int^{\infty}_{0}(\omega (yt/t^ 2)dt\leq C\omega (y)+C\) for all \(y>0.\) The proof is based on the methods introduced by Carleson and Ehrenpreis, Hörmander's solution of the \({\bar \partial}\)-problem, the Phragmén-Lindelöf principle, and the dual characterization of surjective linear maps between Fréchet-Schwartz-spaces.
Hörmander's solution of the \({\bar \partial }\)-problem
dual characterization of surjective linear maps between Fréchet-Schwartz- spaces
scientific article; zbMATH DE number 4118980
scientific article