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BY 4.0 license Open Access Published by De Gruyter Open Access June 6, 2022

Cauchy-Riemann ̄∂-equations with some applications

  • Jie Xiao EMAIL logo and Cheng Yuan
From the journal Complex Manifolds


This paper shows that given 0 < p < 3 and a complex Borel measure µ on the unit disk 𝔻 the inhomogeneous Cauchy-Riemann ̄∂-equation

z¯u(z)=dμ(z)(2πi)-1dz¯dz − a complex Gauss curvature of the weighted disk (𝔻, µ) ᗄ z ∈ 𝔻,

has a distributional solution (initially defined on ̄𝔻 = 𝔻 ∪ 𝕋) u ∈ ℒ2,p(𝕋) (formed of: (i) Morrey’s space M2,0<p<1(𝕋); (ii) John-Nirenberg’s space BMO(𝕋) = 𝒧2,1(𝕋); (iii) Hölder-Lipschitz’s space C C0<p-12<1 (𝕋)), if and only if

𝔻¯z𝔻(1-zw¯)-1dμ¯(w) belongs to the analytic Campanato space ϱ𝒜p(𝔻),

thereby not only extending Carleson’s corona & Wolff’s ideal theorems to the algebra M ϱ𝒜p(𝔻) of all analytic pointwise multiplications of ϱ𝒜p(𝔻), but quadratically generalizing Brownawell’s result on Hilbert’s Nullstellensatz for the analytic polynomial class 𝒫(ℂ).

MSC 2010: 32W05; 11C08; 30H05; 30H80; 46J20; 30H25


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Received: 2022-01-28
Accepted: 2022-05-14
Published Online: 2022-06-06

© 2022 Jie Xiao et al., published by De Gruyter

This work is licensed under the Creative Commons Attribution 4.0 International License.

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