On the characteristics for convolution equations in tube domains

Abstract

We study holomorphic solutions for convolution equations in tube domains. Let ${\mathcal{O}}^{\tau }$ be the sheaf of holomorphic functions in tube domains on the purely imaginary space $\sqrt{-1}{R}^n$ and $\mathcal{S}$ the complex $0\rightarrow \mathscr{O}^{\tau}\rightarrow \mathscr{O}^{\tau}\mu*\rightarrow 0$ generated by the convolution operator $\mu *$ with hyperfunction kernel $\mu$. In this paper, we give a new definition of "the characteristic set" Char$(\mu *)$ using terms of zeros of the total symbol of $\mu *$, and show, under the abstract condition $\left(S\right)$, the equivalence between two notions of characteristics outside of the zero section $T^{*}_{\sqrt{-1}^R^{n}}(\sqrt{-1}R^{n})$. Moreover we conclude that the micro-support SS$\left(\mathcal{S}\right)$ of $\mathcal{S}$ coincides with the characteristics Char$(\mu *)$.