{"title":"柯西黎曼∂方程的一些应用","authors":"J. Xiao, Cheng Yuan","doi":"10.1515/coma-2021-0134","DOIUrl":null,"url":null,"abstract":"Abstract 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 {\\partial _{\\bar z}}u\\left( z \\right) = {{d\\mu \\left( z \\right)} \\over {{{\\left( {2\\pi i} \\right)}^{ - 1}}d\\bar z \\wedge 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 {C^{0 < {{p - 1} \\over 2} < 1}} (𝕋)), if and only if 𝔻¯∋z↦∫𝔻(1-zw¯)-1dμ¯(w) \\mathbb{D} z \\mapsto \\int\\limits_\\mathbb{D} {{{\\left( {1 - z\\bar w} \\right)}^{ - 1}}d\\bar \\mu } \\left( w \\right) 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).","PeriodicalId":42393,"journal":{"name":"Complex Manifolds","volume":"9 1","pages":"170 - 191"},"PeriodicalIF":0.5000,"publicationDate":"2022-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"2","resultStr":"{\"title\":\"Cauchy-Riemann ̄∂-equations with some applications\",\"authors\":\"J. Xiao, Cheng Yuan\",\"doi\":\"10.1515/coma-2021-0134\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract 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 {\\\\partial _{\\\\bar z}}u\\\\left( z \\\\right) = {{d\\\\mu \\\\left( z \\\\right)} \\\\over {{{\\\\left( {2\\\\pi i} \\\\right)}^{ - 1}}d\\\\bar z \\\\wedge 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 {C^{0 < {{p - 1} \\\\over 2} < 1}} (𝕋)), if and only if 𝔻¯∋z↦∫𝔻(1-zw¯)-1dμ¯(w) \\\\mathbb{D} z \\\\mapsto \\\\int\\\\limits_\\\\mathbb{D} {{{\\\\left( {1 - z\\\\bar w} \\\\right)}^{ - 1}}d\\\\bar \\\\mu } \\\\left( w \\\\right) 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).\",\"PeriodicalId\":42393,\"journal\":{\"name\":\"Complex Manifolds\",\"volume\":\"9 1\",\"pages\":\"170 - 191\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2022-01-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"2\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Complex Manifolds\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/coma-2021-0134\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Complex Manifolds","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/coma-2021-0134","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
Cauchy-Riemann ̄∂-equations with some applications
Abstract 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 {\partial _{\bar z}}u\left( z \right) = {{d\mu \left( z \right)} \over {{{\left( {2\pi i} \right)}^{ - 1}}d\bar z \wedge 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
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Complex Manifolds is devoted to the publication of results on these and related topics: Hermitian geometry, Kähler and hyperkähler geometry Calabi-Yau metrics, PDE''s on complex manifolds Generalized complex geometry Deformations of complex structures Twistor theory Geometric flows on complex manifolds Almost complex geometry Quaternionic geometry Geometric theory of analytic functions Holomorphic dynamics Several complex variables Dolbeault cohomology CR geometry.
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