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Ufa Mathematical Journal, 2022, Volume 14, Issue 3, Pages 127–140
DOI: https://doi.org/10.13108/2022-14-3-127
(Mi ufa626)
 

On boundary properties of asymptotically holomorphic functions

A. Sukhovab

a Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevky str. 112, 450008, Ufa, Russia
b Université de Lille, Laboratoire Paul Painlevé, Departement de Mathématique, 59655 Villeneuve d'Ascq, Cedex, France
References:
Abstract: It is well known that for a generic almost complex structure on an almost complex manifold $(M,J)$ all holomorphic (even locally) functions are constants. For this reason the analysis on almost complex manifolds concerns the classes of functions which satisfy the Cauchy-Riemann equations only approximately. The choice of such a condition depends on a considered problem. For example, in the study of zero sets of functions the quasiconformal type conditions are very natural. This was confirmed by the famous work of S. Donaldson. In order to study the boundary properties of classes of functions (on a manifold with boundary) other type of conditions are suitable. In the present paper we prove a Fatou type theorem for bounded functions with $\overline\partial_J$ differential of a controled growth on smoothly bounded domains in an almost complex manifold. The obtained result is new even in the case of $\mathbb{C}^n$ with the standard complex structure. Furthermore, in the case of $\mathbb{C}^n$ we obtain results with optimal regularity assumptions. This generalizes several known results.
Keywords: almost complex manifold, $\overline\partial$-operator, admissible region, Fatou theorem.
Funding agency Grant number
Labex
The author is partially suported by Labex CEMPI.
Received: 28.04.2022
Bibliographic databases:
Document Type: Article
MSC: 32H02, 53C15
Language: English
Original paper language: English
Citation: A. Sukhov, “On boundary properties of asymptotically holomorphic functions”, Ufa Math. J., 14:3 (2022), 127–140
Citation in format AMSBIB
\Bibitem{Suk22}
\by A.~Sukhov
\paper On boundary properties of asymptotically holomorphic functions
\jour Ufa Math. J.
\yr 2022
\vol 14
\issue 3
\pages 127--140
\mathnet{http://mi.mathnet.ru/eng/ufa626}
\crossref{https://doi.org/10.13108/2022-14-3-127}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=4472643}
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  • https://doi.org/10.13108/2022-14-3-127
  • https://www.mathnet.ru/eng/ufa/v14/i3/p131
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    Russian version PDF:47
    English version PDF:16
    References:22
     
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