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


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.


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