Quantitative Carleman-type estimates for holomorphic sections over bounded domains

TL;DR

The paper delivers fully quantitative Carleman-type inequalities for holomorphic sections over bounded domains in $\mathbb{C}^n$, first proving a Sobolev-type inequality for the Laplacian with explicit constants and then extending to holomorphic sections of Hermitian vector bundles under curvature bounds. By tracking constants and exploiting the boundary geometry via $\operatorname{LC}_\Omega$ and $\operatorname{LD}_\Omega$, it provides explicit bounds that translate boundary data into interior $L^{p^*}$ norms for holomorphic objects. The work also furnishes sharp Green-operator and heat-kernel estimates, enabling comprehensive quantitative control of Schwarz kernels and Green forms, and thus producing concrete Carleman-type inequalities with directly usable constants. These results advance quantitative stability and analytic continuation in several complex variables and complex geometry, with potential applications to unique continuation and boundary-value problems for holomorphic sections.

Abstract

This paper establishes quantitative Carleman-type inequalities for holomorphic sections of Hermitian vector bundles over bounded domains in $\mathbb{C}^n$ with $n \geq 2$. 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.

Theorems & Definitions (27)

• Theorem 1.1
• Corollary 1.2
• Corollary 1.3
• Theorem 1.4
• Theorem 1.5
• Lemma 3.1
• proof
• Remark 3.2
• Lemma 3.3
• Lemma 3.4