The topology on the space of \(\delta\)-psh functions in the Cegrell classes
Let \(\Omega\subset\mathbb C^n\) be a bounded hyperconvex domain. Consider the following Cegrell classes of plurisubharmonic functions: \[ \begin{aligned} \mathcal{E}_0:&=\biggl\{\varphi\in\mathcal{PSH}(\Omega)\cap L^\infty(\Omega): \lim_{z\to\partial\Omega}\varphi(z)=0,\;\int_\Omega(dd^c\varphi)^n<+\infty\biggr\},\\ \mathcal{F}:&=\biggl\{\varphi\in\mathcal{PSH}(\Omega): \exists_{(\varphi_j)_{j=1}^\infty\subset\mathcal{E}_0}: \varphi_j\searrow\varphi,\;\sup_j\int_\Omega(dd^c\varphi_j)^n<+\infty\biggr\},\\ \delta\mathcal{F}:&=\mathcal{F}-\mathcal{F},\\ \mathcal{E}:&=\biggl\{\varphi\in\mathcal{PSH}(\Omega): \forall_{x_0\in\Omega}\;\exists_{\omega - \text{a neighborhood of }x_0}\;\exists_{(\varphi_j)_{j=1}^\infty\subset\mathcal{E}_0}:\\ &\hskip120pt \varphi_j\searrow\varphi \text{ on } \omega,\;\sup_j\int_\Omega(dd^c\varphi_j)^n<+\infty\biggr\}, \\ \delta\mathcal{E}:&=\mathcal{E}-\mathcal{E}. \end{aligned} \] Let \[ \begin{aligned} |||u|||:&=\inf\Biggl\{\biggl(\int_\Omega(dd^c(u_1+u_2))^n\biggr)^{1/n}: u_1,u_2\in\mathcal{F},\;u=u_1-u_2\Biggr\},\quad u\in\delta\mathcal{F},\\ \| u\| _K:&=\inf\Biggl\{\biggl(\int_\Omega(dd^c(u_1+u_2))^n\biggr)^{1/n}: u_1,u_2\in\mathcal{F},\;u=u_1-u_2 \text{ on } K\Biggr\},\quad u\in\delta\mathcal{E},\\ U_{K,\varepsilon}:&=\{u\in\mathcal{E}: \| u\| _K<\varepsilon\},\qquad K\Subset\Omega,\;\varepsilon>0. \end{aligned} \] Let \(\tau\) denote the topology generated on \(\delta\mathcal{E}\) by the family of seminorms \(\{\| \;\| _K: K\Subset\Omega\}\), and let \(\tau_1\) be the topology generated on the same space by the basis of neighborhoods of zero \(\{U_{K,\varepsilon}-U_{K,\varepsilon}: K\Subset\Omega\), \(\varepsilon >0\}\). The authors study various topological properties of the spaces \((\delta\mathcal{F},|||\;|||)\), \((\delta\mathcal{E},\tau)\), and \((\delta\mathcal{E},\tau_1)\). In particular, they prove that \((\delta\mathcal{E},\tau_1)\) is a non-separable, non-reflexive Fréchet space.
scientific article; zbMATH DE number 5117889