23/03/18 18:07:24.36 M09HE8oG.net
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URLﾘﾝｸ(arxiv.org)
[Submitted on 12 Dec 2017 (v1), last revised 11 Mar 2018 (this version, v2)]
A proof of Saitoh's conjecture for conjugate Hardy H2 kernels
Qi'an Guan
In this article, we obtain a strict inequality between the conjugate Hardy H2 kernels and the Bergman kernels on planar regular regions with n>1 boundary components, which is a conjecture of Saitoh.
URLﾘﾝｸ(arxiv.org)
P1
When t = w, R?(z, w￣) denotes R?w(z, w￣) for simplicity. When z = w, R?(z) denotes
R?(z, z￣) for simplicity.
Let B(z, w￣) be the Bergman kernel on D. When z = w, B(z) denotes B(z, z￣)
for simplicity.
In [11] (see also [8] and [12]), the following so-called Saitoh’s conjecture was
posed (backgrounds and related results could be referred to Hejhal’s paper [7] and
Fay’s book [4]).
Conjecture 1.1. (Saitoh’s Conjecture) If n > 1, then R?(z) > πB(z).
In the present article, we give a proof of the above Conjecture.
Theorem 1.1. Conjecture 1.1 holds.
One of the ingredients of the present article is using the concavity of minimal L^2
integrations in [5].
Acknowledgements. The author would like to thank Professor Xiangyu , and Professor Fusheng
Deng, Professor Takeo Ohsawa, Professor Saburou Saitoh for helpful discussions
and encouragements. The author would also like to thank the hospitality of Beijing
International Center for Mathematical Research.
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