Holomorphic functions on the complex sphere (Q1109920)

For \(\lambda\),\(\rho\in {\mathbb{C}}\) denote by \(M_{\pi}\) the algebraic subvariety \(\{(z_ 1,...,z_{d+1}):\) \(z^ 2_ 1+...z^ 2_{d+1}=\rho^ 2\}\) of \({\mathbb{C}}^{d+1}\) and put \[ {\mathcal O}_{\lambda}({\mathbb{C}}^{d+1})=\{f\in {\mathcal O}({\mathbb{C}}^{d+1}):\quad (\frac{\partial f}{\partial z_ 1})^ 2+...+(\frac{\partial f}{\partial z_{d+1}})^ 2=-\lambda^ 2f\}. \] The main result of the paper is the following: The restriction map \(F\to F| M_{\rho}\) is a linear topological isomorphism of \({\mathcal O}_{\lambda}({\mathbb{C}}^{d+1})\) onto \({\mathcal O}(M_{\rho})\) if \((\lambda \rho /2)^{-n-(d-1)/2}J_{n+(d- 1)/2}(\lambda \rho)\neq 0,\) for all \(n=\sigma,1,...\); here, \(J_ k\) is the Bessel function of order k. A similar result holds for entire functions of exponential type.