From the Optimal $L^2$-Estimates $(p,n)$ and $(n,p)$-Forms to Uniform Griffiths $p$-Positivity via Convolution and Restriction

Abstract

We prove that the optimal $L^2$-estimate property for $(p,n)$-forms is preserved under standard convolution regularization. As an application, we show that a singular Hermitian metric satisfying the optimal $L^2$-estimate property for $(p,n)$-forms is uniform Griffiths $p$-positive. By establishing the restriction property, we prove that a singular Hermitian metric satisfying the optimal $L^2$-estimate property for $(n,p)$-forms is also uniform Griffiths $p$-positive.