## Osaka Journal of Mathematics

### $L^{2}$-estimates on weakly $q$-convex domains

#### Abstract

We establish an estimate on weakly $q$-convex domains in $\mathbb{C}^{n}$ which provides a unified approach to various existence results for the $\bar{\partial}$-problem. We also prove a Diederich--Fornaess type result for weakly $q$-convex domains.

#### Article information

Source
Osaka J. Math., Volume 52, Number 1 (2015), 1-15.

Dates
First available in Project Euclid: 24 March 2015

https://projecteuclid.org/euclid.ojm/1427202868

Mathematical Reviews number (MathSciNet)
MR3326598

Zentralblatt MATH identifier
1314.32019

#### Citation

Ji, Qingchun; Tan, Guo; Yu, Guangsheng. $L^{2}$-estimates on weakly $q$-convex domains. Osaka J. Math. 52 (2015), no. 1, 1--15. https://projecteuclid.org/euclid.ojm/1427202868

#### References

• H. Ahn and N.Q. Dieu: The Donnelly–Fefferman theorem on $q$-pseudoconvex domains, Osaka J. Math. 46 (2009), 599–610.
• B. Berndtsson: The extension theorem of Ohsawa–Takegoshi and the theorem of Donnelly–Fefferman, Ann. Inst. Fourier (Grenoble) 46 (1996), 1083–1094.
• Z. Błocki: A note on the Hörmander, Donnelly–Fefferman, and Berndtsson $L^{2}$-estimates for the $\overline\partial$-operator, Ann. Polon. Math. 84 (2004), 87–91.
• Z. Błocki: The Bergman metric and the pluricomplex Green function, Trans. Amer. Math. Soc. 357 (2005), 2613–2625.
• Z. Błocki: Suita conjecture and the Ohsawa–Takegoshi extension theorem, to appear in Invent. Math.
• K. Diederich and J.E. Fornaess: Pseudoconvex domains: bounded strictly plurisubharmonic exhaustion functions, Invent. Math. 39 (1977), 129–141.
• K. Diederich and G. Herbort: Extension of holomorphic $L^{2}$-functions with weighted growth conditions, Nagoya Math. J. 126 (1992), 141–157.
• K. Diederich and T. Ohsawa: An estimate for the Bergman distance on pseudoconvex domains, Ann. of Math. (2) 141 (1995), 181–190.
• H. Donnelly and C. Fefferman: $L^{2}$-cohomology and index theorem for the Bergman metric, Ann. of Math. (2) 118 (1983), 593–618.
• L. Hörmander: $L^{2}$ estimates and existence theorems for the $\bar{\partial}$ operator, Acta Math. 113 (1965), 89–152.
• L.-H. Ho: $\overline{\partial}$-problem on weakly $q$-convex domains, Math. Ann. 290 (1991), 3–18.
• Q. Ji, X. Liu and G. Yu: $L^{2}$-estimates on $p$-convex Riemannian manifolds, Adv. Math. 253 (2014), 234–280.
• \begingroup T. Napier and M. Ramachandran: The Bochner–Hartogs dichotomy for weakly $1$-complete Kähler manifolds, Ann. Inst. Fourier (Grenoble) 47 (1997), 1345–1365. \endgroup
• H. Wu: On certain Kähler manifolds which are $q$-complete; in Complex Analysis of Several Variables (Madison, Wis., 1982), Proc. Sympos. Pure Math. 41, Amer. Math. Soc., Providence, RI, 1984, 253–276.