Convolution operators on spaces of real analytic functions

Let \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 operator T μ : 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 when T μ admits a continuous linear right inverse.