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Optimal constant problem in the \(L^2\) extension theorem. (English. French summary) Zbl 1256.32009

The authors show the following result, that gives a final improvement of the constant in the \(L^2\)-extension theorem for weighted integrable holomorphic functions of [T. Ohsawa, Math. Z. 219, No. 2, 215–225 (1995; Zbl 0823.32006)]. They also state and prove a version of Ohsawa’s extension theorem at the level of Stein manifolds, namely:
Theorem 1. Let \(X\) be a Stein manifold of dimension \(n\). Let \(\varphi \) and \(\varphi+\psi\) be holomorphic functions on \(X\). Assume that \(w\) is a holomorphic function on \(X\) such that \(\sup_X\big(\psi + 2 \log |w|\big) \leq 0\) and \(dw\) does not vanish identically on any branch of \(H:=w^{-1}(0)\). Put \(H_0:=\big\{x \in H \,\,|\,\, dw (x)\neq 0\big\}\). Then, for any holomorphic \((n-1)\)-form \(f\) on \(H_0\) satisfying
\[ c_{n-1}\int_{H_0}e^{-(\varphi+\psi)}f \wedge \bar f < \infty, \] where \(c_k:= (-1)^{k(k-1)/2}i^k\), for a positive integer \(k\), there exists a holomorphic \(n\)-form \(F\) on \(X\) satisfying \(F=dw \wedge \widetilde f\) on \(H_0\), with \(\iota^*\widetilde f =f\) (where \(\iota: H \longrightarrow X\) is the inclusion map) and \[ c_n \int_X e^{-\varphi} F\wedge \bar F \leq 2\pi c_{n-1}\int_{H_0}e^{-(\varphi+\psi)}f \wedge \bar f.{(1)} \] In the above-mentioned paper Ohsawa had proved the theorem in case that \(X\) is a pseudoconvex domain in \(\mathbb C^n\) and \(w(z)= z_n\). Instead of estimate (1) he had obtained the weighted estimate
\[ c_n \int_X e^{-\varphi} F\wedge \bar F \leq 2\pi C\cdot c_{n-1}\int_{H_0}e^{-(\varphi+\psi)}f \wedge \bar f{(1')}, \] where \(C\) is some universal constant.
The authors’ concern was to show \((1')\) with \(C=1\).
The present paper contains an application of theorem 1 to the Suita conjecture, which can be stated as follows: Let \(\Omega\) be a Riemann surface with Bergman \((1,0)\)-form \(K_\Omega\) and log-capacity \(c_\Omega\). Then \[ c_\Omega ^2|dz|^2 \leq \pi K_\Omega {(2)} . \] Ohsawa had obtained a first result of the form \[ c_\Omega ^2|dz|^2 \leq 750\pi K_\Omega {(2')} \] for Riemann surfaces. Blocki had shown the Suita conjecture for domains \(\Omega\) in the plane.
Now, the authors prove it in full generality.

MSC:

32F45 Invariant metrics and pseudodistances in several complex variables
30F45 Conformal metrics (hyperbolic, Poincaré, distance functions)

Citations:

Zbl 0823.32006
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References:

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