View source for Ohsawa–Takegoshi L2 extension theorem

{{short description|Result concerning the holomorphic extensions In several complex variables}}
{{DISPLAYTITLE:Ohsawa–Takegoshi ''L''<sup>2</sup> extension theorem}}

In [[several complex variables]], the '''Ohsawa–Takegoshi ''L''<sup>2</sup> extension theorem''' is a fundamental result concerning the [[Holomorphic function#Several variables|holomorphic]] extension of an [[Square-integrable function|<math>L^2</math>]]-holomorphic function defined on a bounded [[Stein manifold]] (such as a [[pseudoconvex]] [[compact (mathematics)|compact]] set in <math>\mathbb{C}^n</math> of dimension less than <math>n</math>) to a domain of higher dimension, with a bound on the growth. It was discovered by [[Takeo Ohsawa]] and Kensho Takegoshi in 1987,<ref>{{harvtxt|Ohsawa|Takegoshi|1987}}</ref> using what have been described as ''ad hoc'' methods involving twisted [[Laplace–Beltrami operator]]s, but simpler proofs have since been discovered.<ref>{{harvtxt|Siu|2011}}</ref> Many generalizations and similar results exist, and are known as theorems of Ohsawa–Takegoshi type.

== See also ==
*[[Suita conjecture]]

==note==
{{Reflist}}

==References==
{{refbegin}}
*{{cite journal |doi=10.1007/s13373-014-0058-2|title=Cauchy–Riemann meet Monge–Ampère|year=2014|last1=Błocki|first1=Zbigniew|journal=Bulletin of Mathematical Sciences|volume=4|issue=3|pages=433–480|s2cid=53582451|doi-access=free|url=http://ruj.uj.edu.pl/bitstreams/e7086bc1-02c4-4461-8297-4b36242ad586/download}}
*{{cite book |doi=10.1007/978-3-0348-8436-5_3|chapter-url=https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/ohsawa_tak.pdf|chapter=On the Ohsawa–Takegoshi–Manivel ''L''<sup>2</sup> extension theorem|title=Complex Analysis and Geometry|year=2000|last1=Demailly|first1=Jean-Pierre|series=Progress in Mathematics |volume=188 |pages=47–82|isbn=978-3-0348-9566-8}}
*{{cite journal |jstor=24523356|last1=Guan|first1=Qi'an|last2=Zhou|first2=Xiangyu|title=A solution of an <math>L^2</math> extension problem with an optimal estimate and applications|journal=Annals of Mathematics|year=2015|volume=181|issue=3|pages=1139–1208|doi=10.4007/annals.2015.181.3.6|s2cid=56205818|arxiv=1310.7169}}
*{{cite journal |doi=10.1007/BF02391775|title=L<sup>2</sup> estimates and existence theorems for the <math>\bar \partial</math> operator |year=1965 |last1=Hörmander |first1=Lars |journal=Acta Mathematica |volume=113 |pages=89–152 |s2cid=120051843 |doi-access=free }}
*{{cite journal | title=On the extension of <math>L^2</math> holomorphic functions | journal=[[Mathematische Zeitschrift]] | year=1987 | volume=195 | issue=2 | pages=197–204 | doi=10.1007/BF01166457|s2cid=122156071 | last1=Ohsawa | first1=Takeo | last2=Takegoshi | first2=Kensho }}
*{{cite journal |doi=10.1142/S0129167X17400055|title=On the extension of <math>L^2</math> holomorphic functions VIII — a remark on a theorem of Guan and Zhou|year=2017|last1=Ohsawa|first1=Takeo|journal=International Journal of Mathematics|volume=28|issue=9}}
*{{cite book |isbn=9784431568513|doi=10.1007/978-4-431-55747-0|title=<math>L^2</math> Approaches in Several Complex Variables: Towards the Oka–Cartan Theory with Precise Bounds|last1=Ohsawa|first1=Takeo|series=Springer Monographs in Mathematics|date=10 December 2018}}
*{{cite book |doi=10.1007/978-981-15-1588-0|title=Bousfield Classes and Ohkawa's Theorem|series=Springer Proceedings in Mathematics & Statistics|year=2020|volume=309|isbn=978-981-15-1587-3|s2cid=242194764|url={{Google books|ssPXDwAAQBAJ|Bousfield Classes and Ohkawa's Theorem|page=426|plainurl=yes}}}}
*{{cite journal | title=Section extension from hyperbolic geometry of punctured disk and holomorphic family of flat bundles | last1= Siu|first1=Yum-Tong | journal=Science China Mathematics |year=2011| volume=54 | issue=8 | pages=1767–1802 | doi=10.1007/s11425-011-4293-7| arxiv=1104.2563 | bibcode=2011ScChA..54.1767S | s2cid=119572640 }}
{{refend}}

==External links==
{{refbegin}}
*{{cite web |title=''L''<sup>2</sup> estimates for the d-bar operator on complex manifolds, Notes de cours, Ecole d'été de Mathématiques (Analyse Complexe), Institut Fourier, Grenoble|date =June 1996|last1=Demailly|first1=Jean-Pierre |url=https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/estimations_l2.pdf}}
*[https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/analmeth_book.pdf Analytic Methods in Algebraic Geometry] ([https://www-fourier.ujf-grenoble.fr/~demailly/documents.html OpenContent book See B5])
{{refend}}

{{DEFAULTSORT:Ohsawa-Takegoshi L2 extension theorem}}
[[Category:Mathematical theorems]]
[[Category:Several complex variables]]