Convolution operators on spaces of real analytic functions (Q2841691)

This paper studies convolution operators on spaces of real analytic functions. Let \(I\subset\mathbb R\) be an open interval, let \(\mu\in A(\mathbb R)'\) and \(G:=\text{conv}(\text{supp}(\mu))\). The author characterizes the surjectivity of the convolution operator \(T_\mu :A(I-G)\rightarrow A(I)\) by means of a new estimate from below for the Fourier transform \(\widehat\mu\), valid on conical subsets of \(\mathbb C\setminus\mathbb R\). Namely, for any \(\eta>0\), there are \(\eta_0\) and a function \(\rho_\eta(t)=o(t)\) such that NEWLINE\[NEWLINE |\widehat\mu(z)|\geq e^{H_G(z)-\eta\text{Im} z}NEWLINE\]NEWLINE if \(\rho_\eta(|\text{Re} z|)\leq \text{Im} z\leq \eta_0|\text{Re} z|\), where \(H_G(z)=\sup_{x\in G}\langle x,\text{Im} z\rangle\). A characterization of when \(T_\mu\) admits a continuous linear right inverse is also considered.NEWLINENEWLINESimilar results, under the hypothesis that \(\text{supp}(\mu)=\{0\}\), were previously proved in [\textit{V. V. Napalkov} and \textit{I. A. Rudakov}, Math. Notes 49, No. 3, 266--271 (1991; Zbl 0763.47012)].