On delta \(m\)-subharmonic functions (Q2836003)

Let \(\Omega\subset \mathbb C^n\) be a bounded \(m\)-hyperconvex domain, i.e, a domain for which there exists a bounded \(m\)-subharmonic exhaustion function, and assume \(1\leq m\leq n\). Let \(\mathcal E_{0,m}\) be the class of bounded \(m\)-subharmonic functions \(u\) in \(\Omega\) with zero boundary values and with bounded total \(m\)-Hessian mass \(\int_{\Omega}\text{H}_m(u)<\infty\). For \(p>0\) let \(\mathcal E_{p,m}\) be the class of negative \(m\)-subharmonic functions \(u\) on \(\Omega\) for which there exists a decreasing sequence \(\{u_j\}\subset \mathcal E_{0,m}\) such that \(\lim_{j\to \infty}u_j=u\) and \(\sup_{j}\int_{\Omega}(-u_j)^p\text{H}_m(u_j)<\infty\). For \(u\in \mathcal E_{p,m}\) let NEWLINE\[NEWLINE e_{p,m}=\int_{\Omega}(-u)^p\text{H}_m(u) NEWLINE\]NEWLINE be the \(p\)-energy of \(u\). The aim of this paper is to prove some properties of the vector space \(\delta \mathcal E_{p,m}=\mathcal E_{p,m}-\mathcal E_{p,m}\). Let \(\succeq\) be the order induced by the cone \(\mathcal E_{p,m}\), i.e., \(u,v\in \delta \mathcal E_{p,m}\) satisfy \(u\succeq v\) if \(u-v\in \mathcal E_{p,m}\).NEWLINENEWLINELet \(\mathcal H_{p,m}\) be the set of all \(m\)-Hessian measures of the functions from \(\mathcal E_{p,m}\), i.e., NEWLINE\[NEWLINE \mathcal H_{p,m}=\{\text{H}_m(u): u\in \mathcal E_{p,m}\}. NEWLINE\]NEWLINE NEWLINENEWLINENEWLINENEWLINE The author proves that the classical order and the order induced by the cone \(\mathcal H_{p,m}\) coincide. Moreover \((\delta E_{p,m},\geq)\) and \((\delta H_{p,m},\geq)\) are Riesz spaces, while \((\delta E_{p,m},\succeq)\) is not a Riesz space but it is a Dedekind \(\sigma\)-complete space. NEWLINENEWLINENEWLINENEWLINE For \(u\in \delta \mathcal E_{p,m}\) one can define the following quasi-norm NEWLINE\[NEWLINE ||u||_{p,m}=\inf\left\{ e_{p,m}(u_1+u_2)^{\frac {1}{p+m}}: u=u_1-u_2, u_1,u_2\in \mathcal E_{p,m}\right\}. NEWLINE\]NEWLINE In the paper it is proved that \((\delta \mathcal E_{p,m},||\cdot||_{p,m})\) is a quasi-Banach space for \(p\neq 1\), and a Banach space for \(p=1\). NEWLINENEWLINENEWLINENEWLINE For \(\mu\in \delta \mathcal H_{p,m}\) one can define the following quasi-norm NEWLINE\[NEWLINE |\mu|_{p,m}=\inf\left\{||u_{\mu_1}||_{p,m}^m+||u_{\mu_2}||_{p,m}^m: \mu=\mu_1-\mu_2, \mu_1,\mu_2\in \mathcal H_{p,m}\right\}, NEWLINE\]NEWLINE where \(u_{\mu_j}\in \mathcal E_{p,m}\) (j=1,2) are the unique solutions to \(\text{H}_m(u_j)=\mu_j\). In the paper it is proved that \((\delta \mathcal H_{p,m},|\cdot|_{p,m})\) is a quasi-Banach space for \(p\neq 1\), and a Banach space for \(p=1\). NEWLINENEWLINENEWLINENEWLINE The main result of the paper is the characterization of the regular (\(X^r\)), order bounded (\(X^b\)) and continuous (\(X'\)) functionals defined on the vector spece \(X\in \{\delta \mathcal E_{p,m}, \delta \mathcal H_{p,m}\}\). Recall that the operator is regular if it can be written as a difference of two positive operators, and it is order bounded if it is bounded on every interval \([x,y]=\{z\in X: y\succeq z\succeq x\}\). NEWLINENEWLINENEWLINENEWLINE The author proves that all three notions of dual space defined above coincide for the vector spaces \(\delta \mathcal E_{p,m}\) and \(\delta \mathcal H_{p,m}\). Moreover for \(p\geq 1\) the space \((\delta \mathcal E_{p,m},\succeq,||\cdot||_{p,m})'\) is the closure of \(\delta \mathcal H_{p,m}\) in the weak\(^*\)-topology \(\sigma((\delta \mathcal E_{p,m})',\delta \mathcal E_{p,m})\). A similar characterization holds for the dual space of \(\delta \mathcal H_{p,m}\) by the space \(\delta \mathcal E_{p,m}\).