A Carleson estimate for the complex Monge-Ampère operator (Q1366624)

Let \(D\) be a pseudoconvex domain in \(\mathbb{C}^n\) that admits a plurisubharmonic defining function \(\rho\) of class \(C^2\). We prove that if \(u_1,\dots,u_r\) are bounded plurisubharmonic functions in \(D\) and \(\omega= dd^c\log 1/(-\rho)\), then \((-\rho)^n dd^cu_1\wedge\dots\wedge dd^cu_r\wedge\omega^{n-r}/(n-r)!\) is a Carleson measure. This is a global variant of the Chern-Levine-Nirenberg inequality.