Solution to the first Cousin problem for vector-valued quasianalytic functions

Debrouwere, Andreas; Vindas, Jasson

doi:10.1007/s10231-017-0649-0

Cited by 7 publications

(11 citation statements)

References 27 publications

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“…for Ω convex this was proven by Rösner [18], whereas for general open sets Ω this was only recently shown by Vindas and the author [5]. Bonet and Domański proved (ii) by viewing -via the Fourier-Laplace transform (Ω is convex!)…”

Section: Introductionmentioning

confidence: 94%

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

Debrouwere

2020

Bulletin des Sciences Mathématiques

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We show that the spaces E {ω} (Ω) of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where ω is a quasianalytic weight function and Ω is an arbitrary open subset of R d . This result was previously shown by Bonet and Domański [2] under the additional assumptions that Ω is convex and ω satisfies the condition (α 1 ). In particular, our work solves Problem 9.7 in [1].2010 Mathematics Subject Classification. 30D60, 46E10, 46A63. Key words and phrases. Spaces of quasianalytic functions of Roumieu type; linear topological invariants for (P LS)-spaces.The author is supported by FWO-Vlaanderen, via the postdoctoral grant 12T0519N.

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Section: Introductionmentioning

confidence: 94%

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

Debrouwere

2020

Bulletin des Sciences Mathématiques

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“…Therefore, being the projective limit of a sequence of LB-spaces it follows that E {ω} (X) is webbed. It has been shown recently by Debrouwere and Vindas [DeVi,Proposition 3.2] that the space of ultradifferentiable functions of Roumieu type is ultrabornological.…”

Section: Spaces Of Ultradifferentiable Functionsmentioning

confidence: 99%

Dynamics of weighted composition operators on function spaces defined by local properties

Kalmes

2019

Studia Math.

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We study topological transitivity/hypercyclicity and topological (weak) mixing for weighted composition operators on locally convex spaces of scalar-valued functions which are defined by local properties. As main application of our general approach we characterize these dynamical properties for weighted composition operators on spaces of ultradifferentiable functions, both of Beurling and Roumieu type, and on spaces of zero solutions of elliptic partial differential equations. Special attention is given to eigenspaces of the Laplace operator and the Cauchy-Riemann operator, respectively. Moreover, we show that our abstract approach unifies existing results which characterize hypercyclicity, resp. topological mixing, of (weighted) composition operators on the space of holomorphic functions on a simply connected domain in the complex plane, on the space of smooth functions on an open subset of R d , as well as results characterizing topological transitiviy of such operators on the space of real analytic functions on an open subset of R d .

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“…Due to the absence of nontrivial compactly supported quasianalytic functions the latter is much more difficult to establish than for Colombeau algebras of non-quasianalytic type. The proof is based on some of our recent results on the solvability of the Cousin problem for vector-valued quasianalytic functions [12]. We mention that the motivation for [12] was precisely this problem.…”

Section: Introductionmentioning

confidence: 99%

“…The proof is based on some of our recent results on the solvability of the Cousin problem for vector-valued quasianalytic functions [12]. We mention that the motivation for [12] was precisely this problem. In Section 5 we embed the infrahyperfunctions of class {M p } into our algebras and show that the multiplication of ultradifferentiable functions of class {M p } is preserved under the embedding we shall construct.…”

Section: Introductionmentioning

confidence: 99%

A nonlinear theory of infrahyperfunctions

Debrouwere¹,

Vernaeve²,

Vindas³

2019

Kyoto J. Math.

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We develop a non-linear theory for infrahyperfunctions (also known as quasianalytic (ultra)distributions [25]). In the hyperfunction case our work can be summarized as follows: We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz's impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve a question raised by M. Oberguggenberger [34, p. 286, Prob. 27.2].

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Solution to the first Cousin problem for vector-valued quasianalytic functions

Abstract: Abstract. We study spaces of vector-valued quasianalytic functions and solve the first Cousin problem in these spaces.

Cited by 7 publications

References 27 publications

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

Dynamics of weighted composition operators on function spaces defined by local properties

A nonlinear theory of infrahyperfunctions

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