Examples of compact sets with non-empty interior which do not admit a continuous linear extension operator for ultradifferentiable functions of Beurling type
Let \(A\) be a closed subset in \(\mathbb{R}^ n\) and \({\mathcal E}_{(\omega)}(\mathbb{R}^ n)\) \(({\mathcal E}_{(\omega)}(A))\) be the class of \(\omega\)-ultradifferentiable functions (Whitney jets) of Beurling type on \(\mathbb{R}^ n\) (on \(A\)), where \(\omega\) is a weight function on \(\mathbb{R}_ +\). A weight function \(\omega\) is called a strong weight function if there is \(C>0\) with \(\int^ \infty_ 1 {\omega(yt)\over t^ 2} dt\leq C(\omega(y)+1)\) for all \(y\geq 0\). \textit{J. Bonet}, \textit{R. W. Braun}, \textit{R. Meise} and \textit{B. A. Taylor} proved that for each closed subset \(A\) in \(\mathbb{R}^ n\) and each \(\omega\)-Whitney jet \(f\in {\mathcal E}_{(\omega)}(A)\) there is \(F\in {\mathcal E}_{(\omega)}(\mathbb{R}^ n)\) such that \(F^ \alpha|_ A= f^ \alpha\), \(\alpha\in \mathbb{N}^ n_ 0\), if and only if \(\omega\) is a strong weight function [Stud. Math. 99, No. 2, 155-184 (1991; Zbl 0738.46009)]. Let \(\omega\) be a weight function and \(A\) be a closed subset in \(\mathbb{R}^ n\). \(A\) is said to have the \(\omega\)-extension property if there is a continuous linear operator \(E_ A: {\mathcal E}_{(\omega)}(A)\to {\mathcal E}_{(\omega)}(\mathbb{R}^ n)\), \(E_ A(f)^ \alpha |_ A= f^ \alpha\) for all \(\alpha\in \mathbb{N}^ n_ 0\), \(f\in {\mathcal E}_{(\omega)}(A)\). \(E_ A\) is called a \(\omega\)- extension operator. In this paper, the author gives examples of fat compact sets having no extension operator in the Beurling classes \({\mathcal E}_{(\omega)}\). The following result is shown: Let \(\omega\) be a strong weight function. Suppose there are \(r\), \(D>1\) such that for each \(T\geq 1\) there is \(t\geq T\) with \(\omega(t)\leq D\log (t)^ r\). Then for each function \(f\in {\mathcal E}_{(\omega)}(\mathbb{R})\) with \(f^ j(0)= 0\), for all \(j\in \mathbb{N}_ 0\), \(K_ f\) does not satisfy the \(\omega\)-extension property, where \(K_ f= \{(x,y)\in [0,1]^ 2: y\leq | f(x)|\}\).
class of \(\omega\)-ultradifferentiable functions of Beurling type
scientific article; zbMATH DE number 550290
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