AbstractLet\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$I\subset \mathbb {R}$\end{document}be an open interval and let\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mu \in A(\mathbb {R})^{\prime }$\end{document}and\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$G:=\mbox{conv}(\mbox{supp}(\mu ))$\end{document}. We characterize the surjectivity of the convolution operatorTμ:A(I−G) →A(I) by means of a new estimate from below for the Fourier transform\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\widehat{\mu }$\end{document}valid on conical subsets of\documentclass{article}\usepackage{amssymb}\begin{document}\pagestyle{empty}$\mathbb {C}\setminus \mathbb {R}$\end{document}. We also characterize whenTμadmits a continuous linear right inverse.