Oka's lemma
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Short description: Theorem in mathematics about plurisubharmonic functions
In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in [math]\displaystyle{ \Complex^n }[/math], the function [math]\displaystyle{ -\log d(z) }[/math] is plurisubharmonic, where [math]\displaystyle{ d }[/math] is the distance to the boundary. This property shows that the domain is pseudoconvex. Historically, this lemma was first shown in the Hartogs domain in the case of two variables, also Oka's lemma is the inverse of the Levi's problem (unramified Riemann domain over [math]\displaystyle{ \Complex^n }[/math]). So maybe that's why Oka called Levi's problem as "problème inverse de Hartogs", and the Levi's problem is occasionally called Hartogs' Inverse Problem.
References
- Harrington, Phillip S. (2007), "A quantitative analysis of Oka's lemma", Mathematische Zeitschrift 256 (1): 113–138, doi:10.1007/s00209-006-0062-7
- Harrington, Phillip S. (2007), "The strong Oka's lemma, bounded plurisubharmonic functions and the [math]\displaystyle{ \overline{\partial} }[/math]-Neumann problem", Asian Journal of Mathematics 11 (1): 127–139, doi:10.4310/AJM.2007.v11.n1.a12
- Herbig, A.-K.; McNeal, J. D. (2012), "Oka's lemma, convexity, and intermediate positivity conditions", Illinois Journal of Mathematics 56 (1): 195–211 (2013), doi:10.1215/ijm/1380287467, https://projecteuclid.org/euclid.ijm/1380287467
- "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur", Japanese Journal of Mathematics 23: 97–155 (1954), 1953, doi:10.4099/jjm1924.23.0_97
- "Pseudoconvexity and the problem of Levi", Bulletin of the American Mathematical Society 84 (4): 481–513, 1978, doi:10.1090/S0002-9904-1978-14483-8
Further reading
- Noguchi, Junjiro (2019). "A brief chronicle of the Levi (Hartog's inverse) problem, coherence and open problem". Notices of the International Congress of Chinese Mathematicians 7 (2): 19–24. doi:10.4310/ICCM.2019.V7.N2.A2.
- Oka, Kiyoshi (1953), "Domaines finis sans point critique intérieur", Japanese Journal of Mathematics 27: 97–155, doi:10.4099/jjm1924.23.0_97, https://www.nara-wu.ac.jp/aic/gdb/nwugdb/oka/ko_ron/f/n09/p001.html PDF TeX
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