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

Let ${\displaystyle {ℇ}_{(\omega )}(\Omega )}$ denote the non-quasianalytic class of Beurling type on an open set Ω in ${\displaystyle {ℝ}^n}$. For ${\displaystyle \mu \in ℇ{\prime }_{(\omega )}\left({ℝ}^n\right)}$ the surjectivity of the convolution operator ${\displaystyle T_{\mu }:{ℇ}_{(\omega )}\left({\Omega }_1\right)\to {ℇ}_{(\omega )}\left({\Omega }_2\right)}$ is characterized by various conditions, e.g. in terms of a convexity property of the pair ${\displaystyle ({\Omega }_1,{\Omega }_2)}$ 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 ${\displaystyle S_{\mu }:D{\prime }_{\{\omega \}}\left({\Omega }_1\right)\to D{\prime }_{\{\omega \}}\left({\Omega }_2\right)}$ between ultradistributions of Roumieu type whenever ${\displaystyle \mu \in ℇ{\prime }_{\{\omega \}}\left({ℝ}^n\right)}$. These results extend classical work of Hörmander on convolution operators between spaces of ${\displaystyle C^{\infty }}$-functions and more recent one of Ciorănescu and Braun, Meise and Vogt.