{"title":"具有有界系数的二重维库阿方程的伯格曼空间","authors":"Víctor A. Vicente-Benítez","doi":"10.1016/j.jmaa.2024.129025","DOIUrl":null,"url":null,"abstract":"<div><div>We develop the theory for the Bergman spaces of generalized <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-solutions of the bicomplex-Vekua equation <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover><mi>W</mi><mo>=</mo><mi>a</mi><mi>W</mi><mo>+</mo><mi>b</mi><mover><mrow><mi>W</mi></mrow><mo>‾</mo></mover></math></span> on bounded domains, where the coefficients <em>a</em> and <em>b</em> are bounded bicomplex-valued functions. We study the completeness of the Bergman space, the regularity of the solutions, and the boundedness of the evaluation functional. For the case <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, the existence of a reproducing kernel is established, along with a representation of the orthogonal projection onto the Bergman space in terms of the obtained reproducing kernel, and an explicit expression for the orthogonal complement. Additionally, we analyze the main Vekua equation (<span><math><mi>a</mi><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>b</mi><mo>=</mo><mfrac><mrow><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover><mi>f</mi></mrow><mrow><mi>f</mi></mrow></mfrac></math></span> with <em>f</em> being a non-vanishing complex-valued function). Results concerning its relationship with a pair of conductivity equations, the construction of metaharmonic conjugates, and the Runge property are presented.</div></div>","PeriodicalId":50147,"journal":{"name":"Journal of Mathematical Analysis and Applications","volume":"543 2","pages":"Article 129025"},"PeriodicalIF":1.2000,"publicationDate":"2024-11-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Bergman spaces for the bicomplex Vekua equation with bounded coefficients\",\"authors\":\"Víctor A. Vicente-Benítez\",\"doi\":\"10.1016/j.jmaa.2024.129025\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<div><div>We develop the theory for the Bergman spaces of generalized <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>p</mi></mrow></msub></math></span>-solutions of the bicomplex-Vekua equation <span><math><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover><mi>W</mi><mo>=</mo><mi>a</mi><mi>W</mi><mo>+</mo><mi>b</mi><mover><mrow><mi>W</mi></mrow><mo>‾</mo></mover></math></span> on bounded domains, where the coefficients <em>a</em> and <em>b</em> are bounded bicomplex-valued functions. We study the completeness of the Bergman space, the regularity of the solutions, and the boundedness of the evaluation functional. For the case <span><math><mi>p</mi><mo>=</mo><mn>2</mn></math></span>, the existence of a reproducing kernel is established, along with a representation of the orthogonal projection onto the Bergman space in terms of the obtained reproducing kernel, and an explicit expression for the orthogonal complement. Additionally, we analyze the main Vekua equation (<span><math><mi>a</mi><mo>=</mo><mn>0</mn></math></span>, <span><math><mi>b</mi><mo>=</mo><mfrac><mrow><mover><mrow><mo>∂</mo></mrow><mo>‾</mo></mover><mi>f</mi></mrow><mrow><mi>f</mi></mrow></mfrac></math></span> with <em>f</em> being a non-vanishing complex-valued function). Results concerning its relationship with a pair of conductivity equations, the construction of metaharmonic conjugates, and the Runge property are presented.</div></div>\",\"PeriodicalId\":50147,\"journal\":{\"name\":\"Journal of Mathematical Analysis and Applications\",\"volume\":\"543 2\",\"pages\":\"Article 129025\"},\"PeriodicalIF\":1.2000,\"publicationDate\":\"2024-11-04\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Mathematical Analysis and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://www.sciencedirect.com/science/article/pii/S0022247X24009478\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q1\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Mathematical Analysis and Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0022247X24009478","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q1","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
摘要
我们发展了有界域上二复-Vekua 方程 ∂‾W=aW+bW‾ 的广义 Lp 解的伯格曼空间理论,其中系数 a 和 b 是有界的二复值函数。我们研究了伯格曼空间的完备性、解的正则性和求值函数的有界性。在 p=2 的情况下,我们确定了重现核的存在,并用得到的重现核表示了伯格曼空间的正交投影,以及正交补码的明确表达式。此外,我们还分析了主 Vekua 方程(a=0, b=∂‾ff,f 为非消失的复值函数)。结果涉及它与一对传导方程的关系、元谐共轭的构造以及 Runge 特性。
Bergman spaces for the bicomplex Vekua equation with bounded coefficients
We develop the theory for the Bergman spaces of generalized -solutions of the bicomplex-Vekua equation on bounded domains, where the coefficients a and b are bounded bicomplex-valued functions. We study the completeness of the Bergman space, the regularity of the solutions, and the boundedness of the evaluation functional. For the case , the existence of a reproducing kernel is established, along with a representation of the orthogonal projection onto the Bergman space in terms of the obtained reproducing kernel, and an explicit expression for the orthogonal complement. Additionally, we analyze the main Vekua equation (, with f being a non-vanishing complex-valued function). Results concerning its relationship with a pair of conductivity equations, the construction of metaharmonic conjugates, and the Runge property are presented.
期刊介绍:
The Journal of Mathematical Analysis and Applications presents papers that treat mathematical analysis and its numerous applications. The journal emphasizes articles devoted to the mathematical treatment of questions arising in physics, chemistry, biology, and engineering, particularly those that stress analytical aspects and novel problems and their solutions.
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