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Abstract

Let $\mu\in \mathcal{A}(\mathbb{R}^{d})'$ be an analytic functional and let $T_\mu$ be the corresponding convolution operator on Sato's space $\mathcal{B}(\mathbb{R}^{d})$ of hyperfunctions. We show that $T_\mu$ is surjective iff $T_\mu$ admits an elementary solution in $\mathcal{B}(\mathbb{R}^{d})$ iff the Fourier transform $\widehat{\mu}$ satisfies Kawai's slowly decreasing condition $(S)$. We also show that there are $0\neq\mu\in \mathcal{A}(\mathbb{R}^{d})'$ such that $T_\mu$ is not surjective on $\mathcal{B}(\mathbb{R}^{d})$.