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1997 | 126 | 2 | 171-198
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On the range of convolution operators on non-quasianalytic ultradifferentiable functions

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Let $ℇ_{(ω)}(Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^n$. For $μ ∈ ℇ'_{(ω)}(ℝ^n)$ 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 $S_μ: D'_{{ω}}(Ω_1) → D'_{{ω}}(Ω_2)$ between ultradistributions of Roumieu type whenever $μ ∈ ℇ'_{{ω}}(ℝ^n)$. 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.
Słowa kluczowe
  • Dpto. Matemática Aplicada, Universidad Politécnica, E-46071 Valencia, Spain
  • Dpto. Análisis Matemático, Universidad de Valencia, E-46100 Burjasot (Valencia), Spain
  • Mathematisches Institut, Heinrich-Heine-Universität, D-40225 Düsseldorf, Fed. Rep. of Germany
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