On the range of convolution operators on non-quasianalytic ultradifferentiable functions

Bonet, Jóse ; Galbis, Antonio ; Meise, R.

Studia Mathematica, Tome 122 (1997), p. 171-198 / Harvested from The Polish Digital Mathematics Library

Résumé

Let $ℇ_{(ω)}(Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^n$. For $μ ∈ ℇ^{\prime }_{(ω)}(ℝ^n)$ the surjectivity of the convolution operator $T_μ: ℇ_{(ω)}(Ω_1) → ℇ_{(ω)}(Ω_2)$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $(Ω_1, Ω_2)$ and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator $S_μ: D^{\prime }_{{ω}}(Ω_1) → D^{\prime }_{{ω}}(Ω_2)$ between ultradistributions of Roumieu type whenever $μ ∈ ℇ^{\prime }_{{ω}}(ℝ^n)$. These results extend classical work of Hörmander on convolution operators between spaces of $C^∞$-functions and more recent one of Ciorănescu and Braun, Meise and Vogt.

Détail

Publié le : 1997-01-01

Zbl 0918.46039

EUDML-ID : urn:eudml:doc:216450

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