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**Projective representations of spaces of quasianalytic functionals.**
*(English)*
Zbl 1062.46022

In [Trans. Am. Math. Soc. 272, 107–160 (1982; Zbl 0599.46026)], K. D. Bierstedt, R. Meise and W. H. Summers considered weighted (LF)-spaces \({\mathcal V}H(G)\) of holomorphic functions on an open set \(G \subset \mathbb{C}^N\) and introduced projective hulls \(H\overline{V}(G)\) of these spaces. \({\mathcal V}H(G)\) is always continuously embedded in \(H\overline{V}(G)\), and \(H\overline{V}(G)\) is a projective limit of Banach spaces. The question whether the (LF)-space coincides algebraically with its projective hull (which is always the case for weighted (LB)-spaces) and whether it carries the topology induced from its projective hull was called the question of projective description or representation.

Let \({\mathcal A}(G)\) denote the space of all complex valued real-analytic functions on an open convex subset \(G\) of \(\mathbb{R}^N\). Its strong dual \({\mathcal A}(G)'_b\) is isomorphic to an (LF)-space \({\mathcal FA}'(G)\) of entire functions on \(\mathbb{C}^N\) via the Fourier-Laplace transform. Each step space of this (LF)-space is a Fréchet space whose topology is given by weighted sup-seminorms. L. Ehrenpreis [Am. J. Math. 82, 522–588 (1960; Zbl 0098.08401)] showed that the topology of \({\mathcal FA}'(\mathbb{R}^N)\) cannot be described by weighted sup-seminorms. J. Bonet and R. Meise [Math. Scand. 94, No. 2, 249–266 (2004; Zbl 1064.46018)] proved that \({\mathcal FA}'(G)\) has a strictly finer topology than its projective hull.

In the present article, the authors prove, however, that \({\mathcal FA}'(G)\) coincides with its projective hull algebraically. The authors also extend this result to the spaces \({\mathcal E}'_{\{\omega\}}(G)\) of quasianalytic functionals, where \({\mathcal E}_{\{\omega\}}(G)\) denotes the space of all \(\omega\)-ultradifferentiable functions of Roumieu type on \(G\) for a given (quasianalytic) weight function \(\omega\). More precisely, they prove that the (LF)-space \({\mathcal FE}'_{\{\omega\}}(G)\) of entire functions on \(\mathbb{C}^N\) which arise as the Fourier-Laplace transforms of the elements of \({\mathcal E}_{\{\omega\}}(G)'_b\) coincides algebraically with its projective hull and has a strictly finer topology. (The latter assertion already follows from the above quoted article of Bonet and Meise.) The corresponding result for non-quasianalytic classes \({\mathcal E}_{\{\omega\}}(G)\) was obtained in [J. Bonet and R. Meise, J. Math. Anal. Appl. 255, 122–136 (2001; Zbl 0960.46027)]. The proof in the present case is different from the one in the non-quasianalytic case.

Let \({\mathcal A}(G)\) denote the space of all complex valued real-analytic functions on an open convex subset \(G\) of \(\mathbb{R}^N\). Its strong dual \({\mathcal A}(G)'_b\) is isomorphic to an (LF)-space \({\mathcal FA}'(G)\) of entire functions on \(\mathbb{C}^N\) via the Fourier-Laplace transform. Each step space of this (LF)-space is a Fréchet space whose topology is given by weighted sup-seminorms. L. Ehrenpreis [Am. J. Math. 82, 522–588 (1960; Zbl 0098.08401)] showed that the topology of \({\mathcal FA}'(\mathbb{R}^N)\) cannot be described by weighted sup-seminorms. J. Bonet and R. Meise [Math. Scand. 94, No. 2, 249–266 (2004; Zbl 1064.46018)] proved that \({\mathcal FA}'(G)\) has a strictly finer topology than its projective hull.

In the present article, the authors prove, however, that \({\mathcal FA}'(G)\) coincides with its projective hull algebraically. The authors also extend this result to the spaces \({\mathcal E}'_{\{\omega\}}(G)\) of quasianalytic functionals, where \({\mathcal E}_{\{\omega\}}(G)\) denotes the space of all \(\omega\)-ultradifferentiable functions of Roumieu type on \(G\) for a given (quasianalytic) weight function \(\omega\). More precisely, they prove that the (LF)-space \({\mathcal FE}'_{\{\omega\}}(G)\) of entire functions on \(\mathbb{C}^N\) which arise as the Fourier-Laplace transforms of the elements of \({\mathcal E}_{\{\omega\}}(G)'_b\) coincides algebraically with its projective hull and has a strictly finer topology. (The latter assertion already follows from the above quoted article of Bonet and Meise.) The corresponding result for non-quasianalytic classes \({\mathcal E}_{\{\omega\}}(G)\) was obtained in [J. Bonet and R. Meise, J. Math. Anal. Appl. 255, 122–136 (2001; Zbl 0960.46027)]. The proof in the present case is different from the one in the non-quasianalytic case.

Reviewer: Klaus Dieter Bierstedt (Paderborn)

### MSC:

46E10 | Topological linear spaces of continuous, differentiable or analytic functions |

46A13 | Spaces defined by inductive or projective limits (LB, LF, etc.) |

46F05 | Topological linear spaces of test functions, distributions and ultradistributions |

46F15 | Hyperfunctions, analytic functionals |