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

Let ${\displaystyle {\epsilon }_{\{\omega \}}\left(I\right)}$ 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 ${\displaystyle \mu \in {\epsilon }_{\{\omega \}}\left(I\right)\prime }$ with ${\displaystyle \supset p(\mu )=\left\{0\right\}}$ one can define the convolution operator ${\displaystyle T_{\mu }:{\epsilon }_{\{\omega \}}\left(I\right)\to {\epsilon }_{\{\omega \}}\left(I\right)}$, ${\displaystyle T_{\mu }\left(f\right)\left(x\right):=⟨\mu ,f(x-·)⟩}$. We give a characterization of the surjectivity of ${\displaystyle T_{\mu }}$ for quasianalytic classes ${\displaystyle {\epsilon }_{\{\omega \}}\left(I\right)}$, 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 ${\displaystyle w\hat{\mu }}$ of μ.