On Whitney's extension theorem for ultradifferentiable functions
The author considers ultradistributions and ultradifferentiable functions of Beurling and of Roumieu type in the sense of Komatsu, assuming on the sequence of positive numbers \((M_n)_n\) the conditions of \(M_0=1\), logarithmic convexity, stability for ultradifferentiable operators and non strong quasi-analyticity. For a nonempty open subset \(\Omega\) of the \(k\)-dimensional Euclidean space \(\mathbb{R}^k\), let \(\mathcal{E}^{(M_n)}(\Omega)\) denote the set of ultradifferentiable functions of Beurling type, \(\mathcal{E}^{\{M_n\}}(\Omega)\) the corresponding set for the Roumieu case and write \(\mathcal{E}^{*}(\Omega)\) when \(*\) is either \((M_n)\) or \(\{M_n\}\). In this setting, the author gives some results related to Whitney's extension theorem and he also gives some structure theorems for ultradistributions with compact support. The following theorems are given in this paper. \noindent Theorem. Let \(M\) be a bounded subset of \(\mathbb{R}^k\) whose closure is \(K\). Let \(F := ( f_\alpha)_{\alpha \in \mathbb{N}_0^k}\) be a jet in \(K\) such that, for each \(h > 0\), there is a constant \(C\) (respectively, there are constants \(h\) and \(C\)) so that \[ | f_\alpha (x)| \leq C \, h^{|\alpha |} M_{|\alpha |}, \;\alpha \in \mathbb{N}_0^k, \;x \in K. \] If \(F\) is a Whitney jet in every compact subset of \(M\), and \(M\) is Whitney-regular, then there is a function f in \(\mathcal{E}^{(M_n)}(\mathbb{R}^k)\) (respectively, \(\mathcal{E}^{\{M_n\}}(\mathbb{R}^k))\) such that \(D^{\alpha} f_{|K} = f_{\alpha}\), \(\alpha \in \mathbb{N}_0^k\). \noindent Theorem. Let \(K\) be a nonempty compact subset of \(\mathbb{R}^k\). If \(K\) is Whitney-regular, then \[ \Phi: \mathcal{E}^{*}(\mathbb{R}^k) \longrightarrow \mathcal{E}^{*}(K) , \Phi(f)=(D^{\alpha} f_{|K})_{\alpha \in \mathbb{N}_0^k} \] is a topological homomorphism. \noindent Theorem. Let \(K\) be a compact subset of \(\Omega\) and let \(T\) be an element of \( \mathcal{E}^{(M_n)'}(\Omega)\) (respectively, \(\mathcal{E}^{\{M_n\}'}(\Omega)\)). If \(K\) is Whitney-regular and \(T\) vanishes strictly in \(\Omega \setminus K\), then there is, for each \(\alpha \in \mathbb{N}_0^k\), a complex Borel measure \(\mu_{\alpha}\) with support contained in \(K\) such that, for some \(h>0\) (respectively, for each \(h>0\)), \[ \sup_{\alpha \in \mathbb{N}_0^k } h^{|\alpha|} M_{|\alpha|} |\mu_{\alpha}| (K) < \infty \qquad \text{and} \qquad T= \sum_{\alpha \in \mathbb{N}_0^k} D^{\alpha} \mu_{\alpha}, \] where the series converges absolutely and uniformly in every bounded subset of \(\mathcal{E}^{*}(\Omega)\).
scientific article; zbMATH DE number 6112713