Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Volume 125 / 1997

Thomas Meyer Studia Mathematica 125 (1997), 101-129 DOI: 10.4064/sm-125-2-101-129


Let $ε_{{ω}}(I)$ denote the space of all ω-ultradifferentiable functions of Roumieu type on an open interval I in ℝ. In the special case ω(t) = t we get the real-analytic functions on I. For $μ ∈ ε_{{ω}}(I)'$ with $supp(μ) = {0}$ one can define the convolution operator $T_μ: ε_{{ω}}(I) → ε_{{ω}}(I)$, $T_μ(f)(x):= ⟨μ,f(x-·)⟩$. We give a characterization of the surjectivity of $T_μ$ for quasianalytic classes $ε_{{ω}}(I)$, where I = ℝ or I is an open, bounded interval in ℝ. This characterization is given in terms of the distribution of zeros of the Fourier Laplace transform $\widehat μ$ of μ.


  • Thomas Meyer

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