$ \bar{\partial} $-equation look at analytic Hilbert's zero-locus theorem

Xiaofen Lv; Jie Xiao; Cheng Yuan; Xiaofen Lv; Jie Xiao; Cheng Yuan

doi:10.3934/era.2022009

Electronic Research Archive

2022, Volume 30, Issue 1: 168-178. doi: 10.3934/era.2022009

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Research article

$ \bar{\partial} $-equation look at analytic Hilbert's zero-locus theorem

1.
Department of Mathematics, Huzhou University, Huzhou, Zhejiang 313000, China
2.
Department of Mathematics and Statistics, Memorial University, St. John's, NL A1C 5S7, Canada
3.
School of Mathematics and Statistics, Guangdong University of Technology, Guangzhou, Guangdong 510520, China

Received: 18 October 2021 Revised: 09 December 2021 Accepted: 09 December 2021 Published: 20 December 2021

Stemming from the Pythagorean Identity $ \sin^2z+\cos^2z = 1 $ and Hörmander's $ L^2 $-solution of the Cauchy-Riemann's equation $ \bar{\partial}u = f $ on $ \mathbb C $, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on $ \mathbb C $ to the quadratic Fock-Sobolev spaces on $ \mathbb C $.
- $ \bar{\partial}u=f $,
- quadratic Fock-Sobolev space,
- analytic Hilbert's Nullstellensatz
Citation: Xiaofen Lv, Jie Xiao, Cheng Yuan. $ \bar{\partial} $-equation look at analytic Hilbert's zero-locus theorem[J]. Electronic Research Archive, 2022, 30(1): 168-178. doi: 10.3934/era.2022009

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Abstract

Stemming from the Pythagorean Identity $ \sin^2z+\cos^2z = 1 $ and Hörmander's $ L^2 $-solution of the Cauchy-Riemann's equation $ \bar{\partial}u = f $ on $ \mathbb C $, this article demonstrates a corona-type principle which exists as a somewhat unexpected extension of the analytic Hilbert's Nullstellensatz on $ \mathbb C $ to the quadratic Fock-Sobolev spaces on $ \mathbb C $.

References

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