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 $p(z) = z_{ 1}^{ 2} + ... + z_{ n}^{ 2} + \lambda (n \geq 2)$.

Categories: math.RT math.FA

Fourier-Borel transformation on the hypersurface of any reduced polynomial