On the \(L^ p\) stability for the complex Monge-Ampère operator
With the standard notation \(d= \partial+ \overline\partial\) and \(\partial^c= i(\overline\partial- \partial)\) we have the complex Monge-Ampère operator \((dd^c)^n\) which for smooth functions takes the form \[ (dd^c)^n= n! 4^n \text{det}\Biggl({\partial^2\over \partial z_j \partial\overline z_k}\Biggr)d\lambda \] (\(d\lambda\) stands for the \(2n\)-dimensional volume form). In fact, one can well define \((dd^c u)^n\) to be positive Borel measure if \(u\) is plurisubharmonic (PSH, for short) and locally bounded near the boundary of a domain where it is defined. Let \(\Omega\) be a bounded hyperconvex domain in \(\mathbb{C}^n\). Take \(F\in C(\overline\Omega)\), \(F\geq 0\), and \(f\in C(\partial\Omega)\) such that there exists \(\chi\in C(\overline\Omega)\cap \text{PSH}(\Omega)\) such that \(\chi|_{\partial\Omega}= f\). If \(\Omega\) is strictly pseudoconvex then there is exactly one solution \(u= u_\Omega(f, F)\) to the following Dirichlet problem: \(u\in C(\overline\Omega)\cap \text{PSH}(\Omega)\), \((dd^c u)^n= Fd\lambda\), \(u|_{\partial\Omega}= f\). We extend this result to any hyperconvex \(\Omega\). This allows us to state the following. Definition. Let \(1\leq p\), \(q\leq \infty\). We say that there is \((p, q)\)-stability in \(\Omega\) if there exists a positive constant \(C\), depending only on \(p\), \(q\), and \(\Omega\) such that for every \(F\in C(\overline\Omega)\) with \(F\geq 0\), one has \[ |u_\Omega(0, F)|_p\leq C|F|^{1/n}_q. \] As we show, the notion of \((p, q)\)-stability is independent of \(\Omega\).
scientific article; zbMATH DE number 838730