Integral representations and uniqueness theorems for entire functions of several variables (Q1189987)

Let \(n\geq 1\), \(1<p<\infty\), \(0<\rho\), \(\sigma<\infty\), \(\alpha>-2n\). Denote by \(H^ p_{\rho,\sigma,\alpha}(\mathbb{C}^ n)\) the space of entire functions \(f\) on \(\mathbb{C}^ n\) such that \[ \| f\|_{p,\rho,\alpha,\sigma}=\int_{\mathbb{C}^ n}| f(z)|^ p.e^{-\sigma| z|^ \rho}\cdot| z|^ \alpha\cdot dm(z)<+\infty \] where \(m\) is the \(2n\)-dimensional Lebesgue measure in \(\mathbb{C}^ n\simeq\mathbb{R}^{2n}\). The main result of the paper asserts that, if \(1<p<\infty\), any \(f\in H^ p_{\rho,\sigma,\alpha}(\mathbb{C}^ n)\) admits an integral representation of the form \[ f(z)=(\rho\cdot\sigma^ \mu/2\pi^ n)\int_{\mathbb{C}^ n}f(w)K(z,w)\cdot e^{-\sigma| w|^ \rho}\cdot| w|^ \alpha dm(w) \] with \(\mu=(2n+\alpha)/\rho\), \[ K(z,w)=\sum^ \infty_{k=0}{\Gamma(k+m)\over\Gamma(k+1)}\cdot{(\sigma^{2/\rho}\langle z,w\rangle)^ k\over\Gamma(\mu+2k/\rho)}. \] Various uniqueness theorems for classes \(H^ p_{\rho,\sigma,\alpha}(\mathbb{C}^ n)\) and construction of functions in \(H^ p_{2,\sigma,0}(\mathbb{C}^ n)\), \(1<p<\infty\), \(0<\sigma<+\infty\) are proposed.