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\(C^k\)-Estimates for \(\bar{\partial }\)-Equation on Certain Convex Domains of Infinite Type in \(\mathbb {C}^n\)

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Abstract

In this paper, \(C^k\)-estimates are obtained for the Henkin solution operator of the Cauchy–Riemann system

$$\begin{aligned} \bar{\partial } u=\varphi \end{aligned}$$

on a class of certain smoothly bounded, convex domains of infinite type in \(\mathbb {C}^n\), where \(\varphi \) is a \(\bar{\partial }\)-closed (0, q)-differential form. It is proved that the Henkin solution of the \(\bar{\partial }\)-equation admits a suitable Hölder gain.

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Acknowledgements

The author would like to thank the referees for useful remarks and comments that led to the improvement of the paper.

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  1. Faculty of Mathematics and Computer Science, University of Science, Vietnam National University Ho Chi Minh City (VNU-HCM), Ho Chi Minh, Vietnam

    Ly Kim Ha

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This research is funded by Vietnam National University HoChiMinh City (VNU-HCM) under Grant Number B2019-18-01. Some parts of the paper were completed during a scientific stay of the author at the Vietnam Institute for Advanced Study in Mathematics (VIASM), whose hospitality is gratefully appreciated.

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Ha, L.K. \(C^k\)-Estimates for \(\bar{\partial }\)-Equation on Certain Convex Domains of Infinite Type in \(\mathbb {C}^n\). J Geom Anal 31, 2058–2087 (2021). https://doi.org/10.1007/s12220-019-00332-x

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  • DOI: https://doi.org/10.1007/s12220-019-00332-x

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