Quantitative Carleman-type estimates for holomorphic sections over bounded domains

This paper establishes quantitative Carleman-type inequalities for holomorphic sections of Hermitian vector bundles over bounded domains in ${\mathbb{C}}^n$ with $n\ge 2$. We first prove a Sobolev-type inequality with explicit constants for the Laplace operator, which leads to quantitative Carleman-type estimates for holomorphic functions. These results are then extended to holomorphic sections of Hermitian vector bundles satisfying certain curvature restrictions, yielding quantitative versions where previously only non-quantitative forms were available. The proofs refine existing methods through careful constant tracking and by estimating the radius of the uniform sphere condition of the boundary through the Lipschitz constant of its outward unit normal vector.