Application and simplified proof of a sharp \(L^2\) extension theorem (Q2788615)

The famous Ohsawa-Takegoshi extension theorem has been recently generalized by several authors. In particular, \textit{Q. Guan} and \textit{X. Zhou} [C. R., Math., Acad. Sci. Paris 350, No. 15--16, 753--756 (2012; Zbl 1256.32009)] proved the existence of extensions of a holomorphic \((n-1)\)-form \(f\) from the regular part of an analytic \((n-1)\)-dimensional subset of a Stein manifold \(X\), \(\dim X =n\), to an \(n\)-form \(F= dw \wedge f\) defined on \(X\) where \(w\) is a given holomorphic function on \(X\). In the present paper this is applied to show the following stability result on the asymptotics of the ratio of the Bergman kernel of a \(C^2\) smooth pseudoconvex domain \(D\) and a weighted Bergman kernel of \(D'\), which is the slice of \(D\) with the hyperplane \(\{ z_n =0 \}\). Assume \(\phi\) is \(C^2\) in a neighbourhood of the closure of \(D\) such that \(\{ \phi >0 \}\) is pseudoconvex and \(D= \{ \phi (z) >|z_n |^2 \} \). Suppose also that \(\phi\) restricted to \(\bar{D' }\) is of the form \(v\delta ^t , \;t>0\), where \(v\) is \(C^2\) smooth and \(\delta\) is the distance from \(\partial D\), \(-\log \phi\) is plurisubharmonic in \(D\), and NEWLINE\[NEWLINE \phi (z) =\phi (z', 0) + o ( \phi (z', 0) + |z_n |^2 ), \;\;z=(z', 0), NEWLINE\]NEWLINE as \(z\) approaches \(\partial D \cap \{ z_n =0 \} \). Let \(K_{D', \phi} (z', w' )\) denote the Bergman kernel of the space of \(L^2\) holomorphic functions on \(D'\) with respect to the measure \(\phi (z', 0) dV_{n-1}\). Then NEWLINE\[NEWLINE \frac{ K_{D', \phi} (z', z' )}{K_{D} ( (z', 0), (z',0) )} \to 1, NEWLINE\]NEWLINE as \(z\) tends to \(\partial D \cap \{ z_n =0 \} .\)NEWLINENEWLINEThe author also proposes some simplification of the proof of Guan and Zhou [loc. cit.].