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 the Levi's problem (unramified Riemann domain over $\mathbb {C} ^{n}$ ). 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, MR 2282262, S2CID 121735220
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), arXiv:1112.5138, doi:10.1215/ijm/1380287467, MR 3117025, S2CID 118437110
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

References

Further reading