Duals to weighted spaces of analytic functions (Q2714024)

Let \(D\) be a bounded convex domain in \(\mathbb C^p\), \(0\in D\), and let \(\{u_n(z)\}\) be a decreasing sequence of convex functions defined on \(D\) such that \(u_n(z)\to\infty\) as \(z\to\partial D\), \(z\in D\). Denote by \(H\) the space of analytic functions on \(D\) satisfying the condition NEWLINE\[NEWLINE |f(z)|e^{-|u_n(z)|}\to 0\quad \text{ for all }n = 1,2,\dots NEWLINE\]NEWLINE as \(\operatorname{dist}(z,\partial D)\to 0\). The space \(H\) is endowed with the topology of inductive limit of \(B\)-spaces. The following theorem by Epifanov describes the image of the dual space \(H^*\) under the Laplace transformation:NEWLINENEWLINENEWLINETheorem (Epifanov). Let \(v_n(\lambda) = \sup_{z\in D}(\operatorname{Re} \langle\lambda,z\rangle - u_n(z))\), \(\lambda\in\mathbb C^p\), \(n = 1,2,\dots\), and let \(P\) be the space of entire functions \(F(\lambda)\) with the topology of inductive limit such that, for some \(n = n(F)\), NEWLINE\[NEWLINE \|F\|_n = \sup_{\lambda\in\mathbb C^p}|F(\lambda)|e^{-v_n(\lambda)} < \infty. NEWLINE\]NEWLINE If the condition NEWLINE\[NEWLINE v_{n+1}(\lambda) \geq v_n(\lambda) + \ln(1 + |\lambda|) + c_n NEWLINE\]NEWLINE holds, then the Laplace transformation \(L\) determines a topology isomorphism from the strong dual space \(H^*\) onto the space \(P\) and, conversely, the operator \(L\) maps isomorphically \(P^*\) onto \(H\).NEWLINENEWLINENEWLINEThe aim of the article is to prove the Epifanov theorem under the assumption that the weighted functions \(v_n\in C^2(\mathbb C^p)\) satisfy the so-called ``accuracy'' growth condition. The proof of this theorem is based on using the Hörmander theorem.