Equicontinuity of families of plurisubharmonic functions with bounds on their Monge-Ampère masses (Q1849431)

Let \(\varOmega\subset\mathbb C^n\) be a strictly pseudoconvex domain. The main results of the paper are the following two theorems. (1) Assume that \(\mathcal{PSH}(\varOmega)\ni u_j\longrightarrow u\in\mathcal{PSH}(\varOmega)\) weakly in \(\varOmega\), \(\liminf_{z\to\partial\varOmega}(u_j-u)(z)\geq 0\), and \((dd^cu_j)^n=f_jdV\), \(j=1,2,\dots\). The author characterizes a class of the functions \(f_j\) such that in the above situation \(u_j\longrightarrow u\) uniformly on \(\varOmega\). As a corollary he gets a result on the equicontinuity of certain families of solutions of the Monge-Ampère equation. (2) Let \(u, v\in\mathcal{PSH}(\varOmega)\), \(\lim_{z\to\partial\varOmega}(u-v)(z)\geq 0\), \((dd^cu)^n=fdV\), \((dd^cv)^n=gdV\) with \(\|f\|_p\), \(\|g\|_p\leq c_p\), \(p>1\), \(\frac 1p+\frac 1q=1\). Then for any \(\alpha\in(0,1)\) there exists \(c'=c'(p,n,\alpha,\varOmega)\) such that \(\|(u-v)_+\|_\infty\leq c'c_p^{\frac{\alpha+1}{n+\alpha n+1}} \|(u-v)_+\|_q^{\frac 1{n+\alpha n+1}}\).