Hyperfunction Fundamental Solutions of Surjective Convolution Operators on Real Analytic Functions

Let μ ∈ A(R)′. The surjectivity of the convolution operator Tμ ≔ μ∗ on real analytic functions is characterized by two equivalent conditions: (1) Tμ admits hyperfunctional elementary solutions E+ (and E−), which are analytic on an angular neighbourhood of ] − ∞, C[ (respectively, ] −C, ∞[) for some C ≤ 0. (2) The Fourier tranform μ̂ of μ satisfies a slowly decreasing condition and the following alternative. There is δ > 0 such that |Im z| ≥ δ |Re z| or |Im z| = o (|z|) for any z with μ̂(z) = 0. (**) (**) was obtained for μ ∈ A({0})′ by Napalkov/Rudakov and Meyer.