Oka's lemma

In mathematics, Oka's lemma, proved by Kiyoshi Oka, states that in a domain of holomorphy in $\mathbb {C} ^{n}$ , the function $-\log d(z)$ is plurisubharmonic, where $d$ 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 Levi's problem (unramified Riemann domain $\mathbb {C} ^{n}$ ). Therefore, Oka himself calls Levi's problem "problème inverse de Hartogs", and 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, MR 2282262
Harrington, Phillip S.; Shaw, Mei-Chi (2007), "The strong Oka's lemma, bounded plurisubharmonic functions and the ${\overline {\partial }}$ -Neumann problem", Asian Journal of Mathematics, 11 (1): 127–139, doi:10.4310/AJM.2007.v11.n1.a12, MR 2304586
Herbig, A.-K.; McNeal, J. D. (2012), "Oka's lemma, convexity, and intermediate positivity conditions", Illinois Journal of Mathematics, 56 (1): 195–211 (2013), MR 3117025
Oka, Kiyoshi (1953), "Sur les fonctions analytiques de plusieurs variables. IX. Domaines finis sans point critique intérieur", Japanese Journal of Mathematics, 23: 97–155 (1954), doi:10.4099/jjm1924.23.0_97, MR 0071089
Siu, Yum-Tong (1978), "Pseudoconvexity and the problem of Levi", Bulletin of the American Mathematical Society, 84 (4): 481–513, doi:10.1090/S0002-9904-1978-14483-8

Oka's lemma

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