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Acta Mathematica Vietnamica

, Volume 44, Issue 2, pp 519–530 | Cite as

On Hölder Estimates with Loss of Order One for the \(\bar {\partial }\) Equation on a Class of Convex Domains of Infinite Type in \(\mathbb {C}^{3}\)

  • Ly Kim HaEmail author
Article
  • 26 Downloads

Abstract

In this paper, we establish a Hölder continuity with loss of order one for the Cauchy-Riemann equation on a class of smoothly bounded, convex domains of infinite type in the sense of Range in \(\mathbb {C}^{3}\). Let Ω be such a domain and let φ be a (0,1)-form defined continuously on \(\bar {\Omega }\). Then, if φ is Lipschitz continuity on bΩ, in the sense of distributions, there exists a function u belonging to a “suitable” Hölder class such that
$$\bar{\partial} u=\varphi \quad \text{ in } {\Omega}. $$

Keywords

\(\bar {\partial }\) Henkin solution operator Hölder continuity Infinite type domains 

Mathematics Subject Classification (2010)

32W05 32F32 32T25 32T99 

Notes

Acknowledgements

The author is grateful to the referee(s) for careful reading of the paper and valuable suggestions and comments.

Funding Information

This research is funded by Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number 101.02-2017.06.

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Copyright information

© Institute of Mathematics, Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

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

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