Abstract
We prove regularity of solutions of the \(\bar{\partial }\)-problem in the Hölder–Zygmund spaces of bounded, strongly \(\mathbf{C}\)-linearly convex domains of class \(C^{1,1}\). The proofs rely on a new analytic characterization of said domains which is of independent interest, and on techniques that were recently developed by the first-named author to prove estimates for the \(\bar{\partial }\)-problem on strongly pseudoconvex domains of class \(C^2\).
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Notes
Sometime simply referred to as “weakly linearly convex”.
While the notions of “strong” and “strict” convexity (resp., “strong” and “strict” \(\mathbf{C}\)-linear convexity) are distinct from one another, there is no distinction between “strong” and “strict” pseudoconvexity and the two terms are often interchanged. See [13, p. 261].
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Acknowledgements
Part of this work was carried out while the second-named author was in residence at the Isaac Newton Institute for Mathematical Sciences during the program Complex Analysis: techniques, applications and computations (EPSRC Grant No. EP/R04604/1). We thank the Institute, and the program organizers, for the generous support and hospitality. Finally, we are grateful to the referee for making several suggestions that have greatly improved the exposition.
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Partially supported by a grant from the Simons Foundation (Award No.: 505027).
Partially supported by the National Science Foundation (DMS 1504589; DMS-1901978) and an INI-Simons fellowship.
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Gong, X., Lanzani, L. Regularity of a \(\overline{\partial }\)-Solution Operator for Strongly \(\mathbf{C}\)-Linearly Convex Domains with Minimal Smoothness. J Geom Anal 31, 6796–6818 (2021). https://doi.org/10.1007/s12220-020-00443-w
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DOI: https://doi.org/10.1007/s12220-020-00443-w