$\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains

TL;DR

The paper addresses obtaining Sobolev-type and trace inequalities for the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, and derives a generalized Sobolev inequality with trace in $\mathbb{R}^n$. It develops a $\bar\partial$-Sobolev inequality using the Bochner-Martinelli formula and Hölder's inequality for three functions, leading to a corollary for real-domain Sobolev-type estimates. Applications include an integral form of the Maximum Modulus Principle for holomorphic functions and an improved $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains, obtained by a weighted Poincaré inequality and the Kohn-Morrey-Hörmander framework. Explicit constants and potential generalizations are discussed, highlighting sharper analytic tools in several complex variables with potential impact on complex PDE and function theory.

Abstract

We prove several Sobolev-type inequalities related to the $\bar\partial$-operator on bounded domains in $\mathbb{C}^n$, which can be viewed as a $\bar\partial$-version of the classical Sobolev inequality and its various generalizations, and apply them to derive a generalization of the Sobolev Inequality with Trace in $\mathbb{R}^n$. As applications to complex analysis, we get an integral form of Maximum Modulus Principle for holomorphic functions, and an improvement of Hörmander's $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains.

Theorems & Definitions (28)

• Theorem 1.1: $L^2$ Poincaré Inequality
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Corollary 1.5
• Theorem 1.6
• Corollary 1.7
• Theorem 1.8
• Corollary 1.9
• Lemma 3.1