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

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

$$\bar{\partial} u=\varphi \quad \text{ in } {\Omega}. $$

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Notes

  1. 1.

    Uniformly total pseudoconvexity is a needed condition for the existence of the holomorphic support function Φ when Ω is non-convex (see [15]).

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Acknowledgements

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

Funding

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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Correspondence to Ly Kim Ha.

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Ha, L.K. On Hölder Estimates with Loss of Order One for the \(\bar {\partial }\) Equation on a Class of Convex Domains of Infinite Type in \(\mathbb {C}^{3}\). Acta Math Vietnam 44, 519–530 (2019). https://doi.org/10.1007/s40306-018-0288-6

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Keywords

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

Mathematics Subject Classification (2010)

  • 32W05
  • 32F32
  • 32T25
  • 32T99