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arXiv · 2401.15597

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

Fusheng Deng·Weiwen Jiang·Xiangsen Qin
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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 H\"ormander's $L^2$-estimate for $\bar\partial$ on bounded strictly pseudoconvex domains.

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Fusheng Deng, Weiwen Jiang, Xiangsen Qin. 2024-01-28. $\bar\partial$ Sobolev-type inequality and an improved $L^2$-estimate of $\bar\partial$ on bounded strictly pseudoconvex domains. https://arxiv.org/abs/2401.15597

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