Fourier-Borel transformation on the hypersurface of any reduced polynomial

Abstract

For a polynomial $p$ on $C^n$, the variety $V_p=\{z\in C^n;p(z)=0\}$ will be considered. Let $\mathrm{Exp}(V_p)$ be the space of entire functions of exponential type on $V_p$, and ${\mathrm{Exp}}^{\prime }(V_p)$ its dual space. We denote by $\partial p$ the differential operator obtained by replacing each variable $z_j$ with $\partial /\partial z_j$ in $p$, and by ${\mathit{O}}_{\partial p}(C^n)$ the space of holomorphic solutions with respect to $\partial p$. When $p$ is a reduced polynomial, we shall prove that the Fourier-Borel transformation yields a topological linear isomorphism: ${\mathrm{Exp}}^{\prime }(V_p)\to {\mathit{O}}_{\partial p}(C^n)$. The result has been shown by Morimoto, Wada and Fujita only for the case $p(z)=z_1^2+\cdots +z_n^2+\lambda (n\ge 2)$.