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In applying the Bochner-Kodaira formula with boundary term to solve the$\bar\partial$ equation with $L^2$ estimates, the gradient term is usually notused. Two potentially important applications of the use of the gradient termare the strong rigidity for holomorphic vector bundles and the very amplenesspart of the Fujita conjecture. In this note we use the gradient term toconstruct holomorphic sections to prove the Thullen-type extension acrosscodimension $1$ for holomorphic vector bundles with Hermitian metric whosecurvature is $L^p$ for some $p>1$. This construction of sections points out atypical way of how the gradient term can be used.

The Role of the Gradient Term of the Bochner-Kodaira Formula in Coherent Sheaf Extension