Analytic extension of smooth functions
Let \(\mathcal E_*(\mathbb{R}^n)\) be one of the spaces \(\mathcal E_{(M_p)}(\mathbb{R}^n)\), \(\mathcal E_{\{M_p\}}(\mathbb{R}^n)\), \(\mathcal E_{(\omega)}(\mathbb{R}^n)\) and \(\mathcal E_{\{\omega\}}(\mathbb{R}^n)\) of ultradifferentiable functions and let \(\mathcal B\mathcal E_*(\mathbb{R}^n)\) be the corresponding space of functions with uniform ultradifferentiable norms on \(\mathbb{R}^n\). \textit{J. Schmets} and \textit{M. Valdivia} [Result. Math. 31, No. 3-4, 374-385 (1997; Zbl 0877.26015) and Math. Nachr. 195, 187-197 (1998; Zbl 0926.26014)] proved the following version for \(\mathcal E_*(\mathbb{R}^n)\) of the classical Whitney extension theorem: If \(F\) is a closed subset of \(\mathbb{R}^n\) then for any \(f\in\mathcal E_*(\mathbb{R}^n)\) there is \(\widetilde f\in\mathcal E_*(\mathbb{R}^n)\) such that \((1)\) \(\widetilde f\) is real analytic on \(\Omega:=\mathbb{R}^n\setminus F\) and \((2)\) \(\partial^{\alpha}\widetilde f_{|F}=\partial^{\alpha}f_{|F}\) for any \(\alpha\in\mathbb{N}^n_0\). Moreover, if \(f\in\mathcal B\mathcal E_*(\mathbb{R}^n)\), \(\widetilde f\) can be obtained by means of a continuous linear operator \(T_*:\;\mathcal B\mathcal E_*(\mathbb{R}^n)\to\mathcal B\mathcal E_*(\mathbb{R}^n)\). In this paper the author gives an alternative, unified proof of the above results based on the following observation: For \(f\in\mathcal C^\infty(\mathbb{R}^n)\), let \(u_f\in D^\prime(\mathbb{R}^{n+1})\) be odd with respect to \(y\), where \((x,y)\in\mathbb{R}^{n+1}\) with \(x\in\mathbb{R}^n\), and such that \(\Delta u_f=2f\otimes\partial_y\delta_y\). Then a solution \(\widetilde f\in\mathcal C^\infty(\mathbb{R}^n)\) of \((1)\) and \((2)\) is given by \(\widetilde f(x)=f(x)\) for \(x\in F\) and \(\widetilde f(x)=u_f(x,h(x))\) for \(x\in\Omega\), where \(h\in\mathcal B\mathcal C^\infty(\mathbb{R}^n)\) is flat on \(F\), real analytic and strictly positive on \(\Omega\). Moreover, if \(f\in\mathcal B\mathcal C^\infty(\mathbb{R}^n)\), we can take \(\widetilde f=T(f)\), where \(T(f)(x):=\frac 2{c_{n+1}}\int f(x-\xi h(x))(1+|\xi|^2)^{-(n+1)/2}d\xi\) for \(x\in\mathbb{R}^n\), where \(c_{n+1}\) is the area of the unit sphere in \(\mathbb{R}^{n+1}\). Thus the solution of the problem \((1)-(2)\) is reduced to the construction of the corresponding function \(h\) which is done by Lemma 2.2 of the paper.
scientific article; zbMATH DE number 1383725
scientific article