Estimates for Monge-Ampère operators acting on positive plurisubharmonic currents (Q1943377)

Let \(\Omega_1\Subset\Omega\) be two open subsets of \({\mathbb C}^N={\mathbb C}^n\times{\mathbb C}^m\). Let \(\varphi: (z,t)\mapsto \varphi(z)\) be a \(C^2\) positive semi-exhaustive plurisubharmonic function on \(\Omega\) such that \(\log\varphi\) is also plurisubharmonic on \(\{\varphi>0\}\). With these notations, we state the main theorem of this paper: Let \(T\) be a positive plurisubharmonic (\(dd^cT\geq 0\)) current of bidegree \((k,k)\), \(1\leq k\leq n\), on \(\Omega\). Let \((\chi_j)_j\) be a family of smooth regularizing kernels which only depend on \(|(z,t)|\) with \((z,t)\in{\mathbb C}^n\times{\mathbb C}^m\), and let \((v_j)_j\) be a decreasing sequence of smooth plurisuharmonic functions converging pointwise to \(\log\varphi\). Assume that the Hausdorff measure \({\mathcal H}_{2(N-k-p)}(\{\varphi=0\})=0\). Then 1) \(T\star\chi_j\wedge(dd^c\log\varphi)^p\) converges weakly in \(\Omega_1\) to a positive current denoted by \(T\wedge(dd^c\log\varphi)^p\). 2) \(T\wedge(dd^cv_j)^p\) converges weakly in \(\Omega_1\) to \(T\wedge(dd^c\log\varphi)^p\). 3) \(T\star\chi_j\wedge(dd^cv_j)^p\) converges weakly in \(\Omega_1\) to \(T\wedge(dd^c\log\varphi)^p\). This theorem is proved by using the Lelong-Skoda potential current associated to a positive closed current and the support theorem for \(\mathbb C\)-flat currents. The case where \(\varphi(z,t)=|z|\) was proved by \textit{H. El Mir} and \textit{I. Feki} [C. R. Acad. Sci., Paris, Sér. I, Math. 327, No. 9, 797--802 (1998; Zbl 0920.32013)]. As an application, the author proves an inequality of Chern-Levine-Nirenberg type for a positive or negative plurisubharmonic current which is defined outside a pluripolar subset of \(\Omega\).