Hyperfunction fundamental solutions of surjective convolution operators on real analytic functions
Let \(\mu\in A(\mathbb{R})\). The author characterizes the surjectivity of the convolution operator \(T_\mu:= \mu*\) on real analytic functions by two equivalent conditions: (1) \(T_\mu\) admits hyperfunctional elementary solutions \(E_+\) (and \(E_-\)), which are analytic on an angular neighbourhood of \(]- \infty, C[\) (respectively, \([- C, \infty[\)) for some \(C\leq 0\). (2) The Fourier transform \(\widehat\mu\) of \(\mu\) satisfies a slowly decreasing condition and the following alternative. There is \(\delta> 0\) such that \[ |\text{Im } z|\geq \delta|\text{Re } z|\quad\text{or}\quad |\text{Im } z|= o(|z|)\quad\text{for any}\quad z\text{ with }\widehat\mu(z)= 0. \] This latter condition was obtained by \(\mu\in A(\{0\})'\) by Napalkov/Rudakov and Meyer.
surjectivity of the convolution operator
scientific article; zbMATH DE number 847459