On Berndtsson's generalization of Prékopa's theorem (Q1771979)

Let \(V\subset\subset {\mathbb C}^n\) be a strictly pseudo-convex domain. Let \(\phi:{\mathbb C}_0\times \overline V\rightarrow {\mathbb R}\) be a smooth plurisubharmonic function such that \(\Phi(w):=\phi(0,w)\) is uniformly plurisubharmonic on \(V\). Assume that the function \[ w\rightarrow \frac{\partial \phi}{\partial z}(0,w) \] is orthogonal in \(L^2(e^{-\Phi})\) to \[ \left\{h\in\,L^2(e^{-\Phi}): \overline\partial h=0\text{ and } \int h(w)e^{-\Phi(w)}\,d\lambda(w)=0\right\}. \] In this paper, the author proves that the function \(\Psi\) which is given by \[ e^{-\Psi(z)}=\int_V e^{-\phi(z,w)}\,d\lambda(w) \] satisfies \(\Delta\Psi(0)\geq 0\). Here \(d\lambda\) is the Lebesgue measure on \({\mathbb C}^n\cong{\mathbb R}^{2n}\).