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
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Notes
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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The author is grateful to the referee(s) for careful reading of the paper and valuable suggestions and comments.
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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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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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DOI: https://doi.org/10.1007/s40306-018-0288-6