Fourier-Borel transformation on the hypersurface of any reduced polynomial

Abstract

For a polynomial p on Cⁿ, the variety V_p={z∈Cⁿ;p(z)=0} will be considered. Let Exp(V_p) be the space of entire functions of exponential type on V_p, and Exp′(V_p) its dual space. We denote by ∂p the differential operator obtained by replacing each variable z_j with ∂⁄∂z_j in p, and by \\mathcal{O}_∂p(Cⁿ) the space of holomorphic solutions with respect to ∂p. When p is a reduced polynomial, we shall prove that the Fourier-Borel transformation yields a topological linear isomorphism: Exp′(V_p)→\\mathcal{O}_∂p(Cⁿ). The result has been shown by Morimoto, Wada and Fujita only for the case p(z)=z₁²+···+z_n²+λ(n≥2).