Locally bounded holomorphic functions and the mixed Hartogs theorem (Q5907089)

Given Fréchet spaces \(E\) and \(F\), with \(D\) an open subset of \(E\), \({\mathcal H}(U,F)\) denotes the space of all holomorphic functions from \(U\) into \(F\) while \({\mathcal H}_{Lb}(U,F)\) denotes the subspace of all holomorphic functions which are bounded on some neighbourhood of each point of \(U\). This paper is concerned with two questions. The first is `Which conditions on Fréchet spaces \(E\) and \(F\) ensure that \({\mathcal H}(U,F)={\mathcal H}_{Lb}(U,F)\) for every open subset of \(E\)?'. The second is `Which conditions on Fréchet spaces \(E\) and \(F\) ensures that separately holomorphic functions on subsets of \(E\times F'\) are holomorphic?' These questions are studied within the classes of Fréchet Schwartz spaces which satisfy various linear topological invariants of Vogt. A Fréchet space \(E\) is said to have property \((LB^\infty)\) if there is a sequence of positive real numbers \((\rho_n)_n\) increasing to \(\infty\) such that for every positive integer \(p\) there is a positive integer \(q\) such that for all \(n_o\) in \({\mathbb N}\) there is \(N_o\geq n_o\), \(C>0\) and a positive integer \(k\) with \(n_o\leq k\leq N_o\) such that \[ \| u\| _q^{*1+\rho_k}\leq C\| u\| _k^*\| u\| _p^{*\rho_k} \] for all \(u\) in \(E'\). A Fréchet space \(E\) with fundamental system of seminorms \((\| \cdot\| _k)_k\) is said to be a DN-space if there is a seminorm \(\| \cdot\| \) on \(E\) such that for any positive integer \(k\) there is a positive integer \(n\) and \(C>0\) such that \[ \| x\| _k\leq r\| x\| +\tfrac Cr\| x\| _{k+n} \] for all \(r>0\) and all \(x\in E\). Finally, a Fréchet space \(E\) with fundamental system of seminorms \((\| \cdot\| _k)_k\) is said to have property \((\widetilde\Omega)\) if for every positive integer \(p\) there is a positive integer \(q\) and \(d>0\) such that for any positive integer \(k\) there is \(C>0\) such that \[ \| u\| _q^{*(1+d)}\leq C\| u\| _k^*\| u\| _{p}^* \] for all \(u\) in \(E'\). The authors prove that when \(U\) is open in a Fréchet nuclear space \(E\) with \((\widetilde\Omega)\) and \(F\) is a Fréchet space with \((DN)\) then \({\mathcal H}(U,F)={\mathcal H}_{Lb}(U,F)\). The equality \({\mathcal H}(U,F)={\mathcal H}_{Lb}(U,F)\) also holds for every open subset \(U\) of \(E\times B\) when \(B\) is a Banach space, \(E\) is a Fréchet nuclear space with \((\widetilde\Omega)\) and \(F\) has property \((LB^\infty)\). Using these results, the authors proceed to show that every separately holomorphic function on \(U\times F'\) is holomorphic when \(U\) is open in a Fréchet nuclear space with \((\widetilde\Omega)\) and \(F\) is a Fréchet Schwartz space with \((DN)\). Every separately holomorphic function on \(E\times U\) is also shown to be holomorphic when \(E\) is a Fréchet nuclear space with \((\overline{\overline{\Omega}})\) and \(U\) is an open subset of a Fréchet Schwartz space with \((DN)\). Examples of pairs of Fréchet spaces where there are holomorphic functions which are not locally bounded are also given. The paper contains considerably many misprints.