The first main result is a characterization of Nakano positivity of Riemannian vector bundles over bounded domains in terms of solvability of the d-equation with certain good $L^2$ estimate condition. As an application, we give an alternative proof of the matrix-valued Prekopa's theorem that is originally proved by Raufi. The second main result contains a characterization of convexity of smoothly bounded domains in ${\mathbb{R}}^n$ in terms $L^2$ estimate condition for the d-equation, and a characterization of pseudoconvexity of smoothly bounded domains in ${\mathbb{C}}^n$ in terms $L^2$ estimate condition for the $\overline{\partial }$-equation. Our methods are inspired by the recent works of Deng-Ning-Wang-Zhou on characterization of Nakano positivity of Hermitian holomorphic vector bundles and positivity of direct image sheaves associated to holomorphic fibrations.

第一个主要结果是在d方程的可解性方面描述了有界域上黎曼向量丛的 Nakano 正性${升}^2$估计条件。作为应用，我们给出了最初由 Raufi 证明的矩阵值 Prekopa 定理的另一种证明。第二个主要结果包含平滑有界域的凸性特征${\mathbb{电阻}}^n$ 就 ${升}^2$d方程的估计条件，以及平滑有界域的伪凸性的表征${\mathbb{C}}^n$ 就 ${升}^2$ 估计条件 $\overline{\partial }$-方程。我们的方法受到 Deng-Ning-Wang-Zhou 最近关于 Hermitian 全纯向量丛的 Nakano 正性表征和与全纯纤维化相关的直接图像层的正性的研究的启发。