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Whitney’s extension theorem for nonquasianalytic classes of ultradifferentiable functions. (English) Zbl 0738.46009
For a weight function \(\omega\) (on \(\mathbb{R}_ +\)) let \({\mathcal E}_{(\omega)}(\mathbb{R}^ N)\) resp. \({\mathcal E}_{\{\omega\}}(\mathbb{R}^ N)\) denote the nonquasi-analytic class of \(\omega\)-ultradifferntiable functions of Beurling resp. Roumieu type. If a result holds for both types, we will write \({\mathcal E}_ *\). R. Meise, B. A. Taylor [Ark. Mat. 26, No. 2, 265-287 (1988; Zbl 0683.46020)] and J. Bonet, R. Meise, B. A. Taylor [Proc. R. Ir. Acad., Sect. A 89, No. 1, 53-66 (1989; Zbl 0654.46029)] characterized those weight functions \(\omega\), here called strong weight functions, for which an analog of E. Borel’s theorem holds for \({\mathcal E}_{(\omega)}(\mathbb{R}^ N)\) and \({\mathcal E}_{\{\omega\}}(\mathbb{R}^ N)\), respectively; i.e., such that for each family \((a_ \alpha)_ \alpha\) of complex numbers in the natural sequence space associated with \({\mathcal E}_ *\), there exists \(f\in {\mathcal E}_ *\) such that \(f^{(\alpha)}(0)=a_ \alpha\) for all \(\alpha\in \mathbb{N}_ 0^ N\).
In the present article, the authors prove that for strong weight functions \(\omega\) and arbitrary nonvoid closed sets \(A\subset\mathbb{R}^ N\) even the analog of Whitney’s extension theorem holds; i.e., the restriction map of \({\mathcal E}_ *\) to the space of Whitney jets of type \({\mathcal E}_ *\) on \(A\) is surjective. This extends results of J. Bruna, S. Y. Chung and D. Kim, J.-M. Kantor, and H. Komatsu. In the Roumieu case, the proof is a modification of the one by J. Bruna [J. Lond. Math. Soc., II. Ser. 22, 495-505 (1980; Zbl 0419.26010)], but the existence of appropriate cut-off functions is now reduced — by Hörmander’s solution of the \(\overline{\partial}\)-problem — to the existence of subharmonic functions with special properties. The Beurling case is then reduced to the Roumieu case. A simpler description of Whitney jets of type \({\mathcal E}_ *\) follows if the closed set \(A\) satisfies \(A=\overline{\text{Å}}\) and has Whitney’s property \((P)\).

MSC:
46E25 Rings and algebras of continuous, differentiable or analytic functions
46E10 Topological linear spaces of continuous, differentiable or analytic functions
46F05 Topological linear spaces of test functions, distributions and ultradistributions
58C25 Differentiable maps on manifolds
26E10 \(C^\infty\)-functions, quasi-analytic functions
30D15 Special classes of entire functions of one complex variable and growth estimates
31A05 Harmonic, subharmonic, superharmonic functions in two dimensions
32U05 Plurisubharmonic functions and generalizations
32W05 \(\overline\partial\) and \(\overline\partial\)-Neumann operators
42B10 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
46A04 Locally convex Fréchet spaces and (DF)-spaces
46A11 Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)
46A13 Spaces defined by inductive or projective limits (LB, LF, etc.)
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