Linear extension operators for ultradifferentiable functions of Beurling type on compact sets. (English) Zbl 0696.46001

Meise, Reinhold; Taylor, B. Alan

Am. J. Math. 111, No. 2, 309-337 (1989).

Let \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\) denote the space of \(\omega\)- ultradifferentiable functions on \({\mathbb{R}}^ n\) in the sense of Beurling and Björck, and let \({\mathcal E}_{\omega}(K)\) denote the space of Whitney fields on a compact set \(K\subseteq {\mathbb{R}}^ n\) in this class. A main result of this interesting article is the following
Theorem. Let \(\omega\) satisfy this condition of strong non-quasi- analyticity: \[ \exists C>0\quad \forall y>0:\quad \int^{\infty}_{1}(\omega (yt)/t^ 2)dt\leq C(\omega (y)+1). \] Let \(K\subseteq {\mathbb{R}}^ n\) be compact and of the form \(K=\prod^{m}_{j=1}\bar G_ j,\) where \(G_ j\subseteq {\mathbb{R}}^{n_ j}\) are open with real analytic boundary. Then the restriction map \(\rho\) : \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\to {\mathcal E}_{\omega}(K)\) admits a continuous linear right inverse.
The proof is based on a splitting theorem of D. Vogt and M. J. Wagner [Studia Math. 67, 225-240 (1980; Zbl 0464.46010)] for short exact sequences of nuclear Fréchet spaces. The authors showed in [Math. Nachr. 142, 45-72 (1989)] that \({\mathcal E}_{\omega}(K)\) has property (DN), and here they prove that the spaces \({\mathcal D}_{\omega}(\bar G)\) of functions in \({\mathcal E}_{\omega}({\mathbb{R}}^ n)\) with compact support in \(\bar G,\) have property (\(\Omega)\) where \(G\subseteq {\mathbb{R}}^ n\) is open and bounded with real analytic boundary. This proof, via Fourier transform, uses a decomposition lemma from the authors’ article [Math. Nachr. 142, 45-72 (1989)] and requires a careful analysis of the behaviour of the harmonic extension of certain functions on the real line.
Using variations of these methods of proof, for compact sets \(K\subseteq {\mathbb{R}}^ n\) as in the theorem, the authors then prove the isomorphisms \(D_{\omega}(K)\cong \Lambda_{\infty}(\Omega (j^{1/N}))\) for the Beurling case, and \(D_{\{\omega \}}(K)\cong \Lambda_ 1(\omega (j^{1/N}))'_ b\) for the space of \(\omega\)-ultradifferentiable functions of Roumieŭ type with support in K. Under an additional condition on \(\omega\), the authors also construct a continuous linear right inverse for the Borel restriction map \(\rho: {\mathcal E}_{\omega}({\mathbb{R}}^ N)\to {\mathcal E}_{\omega}\{0\}.\)

Reviewer: W.Kaballo

MSC:

46A22	Theorems of Hahn-Banach type; extension and lifting of functionals and operators
46F05	Topological linear spaces of test functions, distributions and ultradistributions
46A13	Spaces defined by inductive or projective limits (LB, LF, etc.)
46A11	Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.)

Keywords:

space of \(\omega\)-ultradifferentiable functions; space of Whitney fields on a compact set; condition of strong non-quasi-analyticity; restriction map; continuous linear right inverse; splitting theorem; Beurling case; \(\omega\)-ultradifferentiable functions of Roumieŭ type; Borel restriction

Citations:

Zbl 0464.46010