Convolution Regularization Preserves the $L^2$-Estimate Property for $(1,n)$-Forms

Abstract

In this paper, we prove that the \(L^2\)-estimate property for \((1,n)\)-forms is preserved under the standard convolution regularization. As applications, we show that any singular Hermitian metric satisfying the optimal or multiple coarse \(L^2\)-estimate property for \((n,1)\) or \((1,n)\)-forms is Griffiths semi-positive. This resolves a question posed by Deng--Ning--Wang and a question by Inayama.

arXiv commentThe proof of Theorem 1.4 began with an incomplete AI-generated argument for planar domains. Examining this flawed work led me to results of Raufi and Watanabe on Nakano negative singular metrics, motivating a generalization to higher dimensions. Together with the results in the first version of this paper, we answer affirmatively questions by Deng-Ning-Wang and Inayama