On a family of quasimetric spaces in generalized potential theory. (English) Zbl 1495.32082
Let \(n\geq2\), \(1\leq m\leq n\), \(p>0\), and let \(\Omega\subset\mathbb C^n\) be an \(m\)-hyperconvex domain. Let \(\mathcal E_{p,m}(\Omega)\) denote Cegrell’s generalized energy class. Put
\[\boldsymbol{J}_p(u,v):=\left(\int|u-v|^p(H_m(u)+H_m(v))\right)^{\frac1{p+m}}, \quad u,v\in\mathcal E_{p.m}(\Omega),\]
where \(H_m\) stands for the \(m\)-Hessian operator. Let \(\mathcal M_{p,m}:=\{\mu: \mu\) is a non-negative Radon measure on \(\Omega\) such that \(H_m(u)=\mu\) for some \(u\in\mathcal E_{p,m}(\Omega)\}\). The main results of the paper are the following theorems:
– \((\mathcal E_{p,m}(\Omega), \boldsymbol{J}_p)\) is a complete quasimetric space.
– Let \(\mu\in\mathcal M_{p,m}\), \(0\leq f\), \(f_j\leq1\) be measurable functions such that \(f_j\longrightarrow f\) in \(L^1_{\mathrm{loc}}(\mu)\). Then \(\boldsymbol{J}_p(U(f_j\mu), U(f\mu))\longrightarrow0\), where \(u=U(\nu)\in\mathcal E_{p,m}(\Omega)\) stands for the unique solution to the Dirichlet problem \(H_m(u)=\nu\).
The authors discuss also the case of connected compact Kähler manifolds and generalize the above results to that case.
– \((\mathcal E_{p,m}(\Omega), \boldsymbol{J}_p)\) is a complete quasimetric space.
– Let \(\mu\in\mathcal M_{p,m}\), \(0\leq f\), \(f_j\leq1\) be measurable functions such that \(f_j\longrightarrow f\) in \(L^1_{\mathrm{loc}}(\mu)\). Then \(\boldsymbol{J}_p(U(f_j\mu), U(f\mu))\longrightarrow0\), where \(u=U(\nu)\in\mathcal E_{p,m}(\Omega)\) stands for the unique solution to the Dirichlet problem \(H_m(u)=\nu\).
The authors discuss also the case of connected compact Kähler manifolds and generalize the above results to that case.
Reviewer: Marek Jarnicki (Kraków)
MSC:
| 32U05 | Plurisubharmonic functions and generalizations |
| 31C45 | Other generalizations (nonlinear potential theory, etc.) |
| 31E05 | Potential theory on fractals and metric spaces |
| 46E36 | Sobolev (and similar kinds of) spaces of functions on metric spaces; analysis on metric spaces |
Keywords:
Caffarelli-Nirenberg-Spruck model; Cegrell class; complex Hessian operator; \(m\)-subharmonic function; quasimetric space; stabilityReferences:
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