Fourier-Borel transformation on the hypersurface of any reduced polynomial

Abstract

For any polynomial $p$ on $\mathbf{C}^{n}$, a variety $ V_{p} = \{z \in \mathbf{C}^{n} ; p(z)=0 \} $ will be considered. Let $\text{Exp}(V_{p})$ be the space of holomorphic functions of expotential growth on $V_{p}$. We shall prove that the Fourier-Borel transformation yields an isomorphism of the dual space $\text{Exp}'(V_{p})$ with the space of holomorphic solutions $\mathcal{O}_{\partial p}(\mathbf{C}^{n})$ with respect to the differential operator $\partial p$ which is obtained by replacing each variable $z_{j}$ with $\partial / \partial z_{j}$ in $p$ when $p$ is a reduced polynomial. The result has been shown by Morimoto and by Morimoto-Wada-Fujita only for the case $p(z) = z_{1}^{2} + ... + z_{n}^{2} + \lambda (n \geq 2)$.