A solution of an $L^{2}$ extension problem with an optimal estimate and applications

Abstract

In this paper, we prove an $L^2$ extension theorem with an optimal estimate in a precise way, which implies optimal estimate versions of various well-known $L^2$ extension theorems. As applications, we give proofs of a conjecture of Suita on the equality condition in Suita’s conjecture, the so-called $L$-conjecture, and the extended Suita conjecture. As other applications, we give affirmative answer to a question by Ohsawa about limiting case for the extension operators between the weighted Bergman spaces, and we present a relation of our result to Berndtsson’s important result on log-plurisubharmonicity of the Bergman kernel.

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      author = {Ohsawa, Takeo},
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      ZBLNUMBER = {1111.32006},
      }
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    @incollection{zhou-zhu2012,
      author = {Zhou, Xiangyu and Zhu, L. F.},
      TITLE = {${L}^2$-extension theorem: revisited},
      BOOKTITLE={Fifth International Congress of Chinese Mathematicians. Part 1, 2},
      SERIES={AMS/IP Stud. Adv. Math.},
      VOLUME={51},
      PUBLISHER={Amer. Math. Soc.},
      ADDRESS={Providence, RI},
      YEAR={2012},
      MRNUMBER = {2908088},
      ZBLNUMBER = {1250.32008},
      }
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    @book {grauert-remmert1, MRKEY = {0580152},
      AUTHOR = {Grauert, Hans and Remmert, Reinhold},
      TITLE = {Theory of {S}tein Spaces},
      SERIES = {Grundl. Math. Wissen.},
      VOLUME = {236},
      NOTE = {translated from the German by Alan Huckleberry},
      PUBLISHER = {Springer-Verlag},
      YEAR = {1979},
      PAGES = {xxi+249},
      ISBN = {3-540-90388-7},
      MRCLASS = {32-01},
      MRNUMBER = {0580152},
      ADDRESS = {New York},
      ZBLNUMBER = {0433.32007},
      DOI = {10.1007/978-1-4757-4357-9},
      }

Authors

Qi'an Guan

Beijing International Center for Mathematical Research, and School of Mathematical Sciences, Peking University, Beijing 100871, China

Xiangyu Zhou

Institute of Mathematics, AMSS, and Hua Loo-Keng Key Laboratory of Mathematics, Chinese Academy of Sciences, Beijing 100190, China