Regularity and comparison principles in complex analysis and locally integrable structures
Autor(a) principal: | |
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Data de Publicação: | 2024 |
Tipo de documento: | Tese |
Idioma: | eng |
Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
Texto Completo: | https://www.teses.usp.br/teses/disponiveis/45/45132/tde-21022024-192259/ |
Resumo: | This thesis deals with three topics, related to complex analysis in several variables, CR geometry and systems of partial differential equations. Firstly, we study a comparison principle between Levi-flat CR structures and systems of real vector fields. In a family of models given by fibre bundles with complex fiber, we compute completely the cohomology of the associated differential complex. We also introduce some compact models (parametrized by a certain number of forms of type (0, 1)) and, in this case, study questions of global hypoellipticity and prove, in several cases, a comparison isomorphism between the cohomology spaces. In the second part, we study (in the one dimensional case) the Fréchet algebra A^\\infty(K) (introduced recently by Cordaro, Della Sala and Lamel, to treat questions about the Borel map). We prove some function-theoretic properties about this algebra (like a localization property, in analogy with the uniform case P(K)) and study the question of determining when this algebra coincide with the full algebra of formal power series with coefficients in C(K). To do this, we introduce as the main tool a generalization of the Cauchy transform. Finally, we study a problem of regularizability of locally integrable structures. Inspired by a recent result of Kossovskiy and Zaitsev, we present a family of structures (of non-CR type) and a sufficent condition that guarantees when such structures are equivalent to real-analytic ones. |
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Regularity and comparison principles in complex analysis and locally integrable structuresRegularidade e princípios de comparação em análise complexa e estruturas localmente integráveisCR geometryEstruturas localmente integráveisGeometria CRGlobal hypoellipticityHipoelipticidade globalLocally integrable structuresThis thesis deals with three topics, related to complex analysis in several variables, CR geometry and systems of partial differential equations. Firstly, we study a comparison principle between Levi-flat CR structures and systems of real vector fields. In a family of models given by fibre bundles with complex fiber, we compute completely the cohomology of the associated differential complex. We also introduce some compact models (parametrized by a certain number of forms of type (0, 1)) and, in this case, study questions of global hypoellipticity and prove, in several cases, a comparison isomorphism between the cohomology spaces. In the second part, we study (in the one dimensional case) the Fréchet algebra A^\\infty(K) (introduced recently by Cordaro, Della Sala and Lamel, to treat questions about the Borel map). We prove some function-theoretic properties about this algebra (like a localization property, in analogy with the uniform case P(K)) and study the question of determining when this algebra coincide with the full algebra of formal power series with coefficients in C(K). To do this, we introduce as the main tool a generalization of the Cauchy transform. Finally, we study a problem of regularizability of locally integrable structures. Inspired by a recent result of Kossovskiy and Zaitsev, we present a family of structures (of non-CR type) and a sufficent condition that guarantees when such structures are equivalent to real-analytic ones.Esta tese trata de três tópicos, relacionados a análise complexa em várias variáveis, geometria CR e sistemas de equações diferenciais parciais. Primeiramente, estudamos um princípio de comparação entre estruturas CR Levi-flat e sistemas de campos vetoriais reais. Em uma família de modelos dados por fibrados com fibra complexa, calculamos completamente a cohomologia do complexo diferencial associado. Introduzimos também alguns modelos compactos (parametrizados por um certo número de formas de tipo (0, 1)), e neste caso, estudamos questões de hipoelipticidade global e provamos, em vários casos, um isomorfismo de comparação entre os espaços de cohomologia. Na segunda parte, estudamos (no caso unidimensional) a álgebra de Fréchet A^\\infty(K) (introduzida recentemente por Cordaro, Della Sala e Lamel, para tratar de questões como a aplicação de Borel). Provamos algumas propriedades desta álgebra (como uma propriedade de localização, análoga ao caso uniforme P(K)) e estudamos a questão de determinar quando esta álgebra coincide com o espaço das séries formais com coeficientes em C(K). Para tal, introduzimos como ferramenta fundamental uma generalização da transformada de Cauchy. Finalmente, estudamos uma questão a respeito da regularização de estruturas localmente integráveis. Inspirados em um resultado recente de Kossovskiy e Zaitsev, apresentamos uma família de estruturas (de tipo não-CR) e uma condição suficiente que garante quando tais estruturas são equivalentes a estruturas real-analíticas.Biblioteca Digitais de Teses e Dissertações da USPCordaro, Paulo DomingosSilva, Vinícius Novelli da2024-02-05info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttps://www.teses.usp.br/teses/disponiveis/45/45132/tde-21022024-192259/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2024-02-27T20:43:03Zoai:teses.usp.br:tde-21022024-192259Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212024-02-27T20:43:03Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
dc.title.none.fl_str_mv |
Regularity and comparison principles in complex analysis and locally integrable structures Regularidade e princípios de comparação em análise complexa e estruturas localmente integráveis |
title |
Regularity and comparison principles in complex analysis and locally integrable structures |
spellingShingle |
Regularity and comparison principles in complex analysis and locally integrable structures Silva, Vinícius Novelli da CR geometry Estruturas localmente integráveis Geometria CR Global hypoellipticity Hipoelipticidade global Locally integrable structures |
title_short |
Regularity and comparison principles in complex analysis and locally integrable structures |
title_full |
Regularity and comparison principles in complex analysis and locally integrable structures |
title_fullStr |
Regularity and comparison principles in complex analysis and locally integrable structures |
title_full_unstemmed |
Regularity and comparison principles in complex analysis and locally integrable structures |
title_sort |
Regularity and comparison principles in complex analysis and locally integrable structures |
author |
Silva, Vinícius Novelli da |
author_facet |
Silva, Vinícius Novelli da |
author_role |
author |
dc.contributor.none.fl_str_mv |
Cordaro, Paulo Domingos |
dc.contributor.author.fl_str_mv |
Silva, Vinícius Novelli da |
dc.subject.por.fl_str_mv |
CR geometry Estruturas localmente integráveis Geometria CR Global hypoellipticity Hipoelipticidade global Locally integrable structures |
topic |
CR geometry Estruturas localmente integráveis Geometria CR Global hypoellipticity Hipoelipticidade global Locally integrable structures |
description |
This thesis deals with three topics, related to complex analysis in several variables, CR geometry and systems of partial differential equations. Firstly, we study a comparison principle between Levi-flat CR structures and systems of real vector fields. In a family of models given by fibre bundles with complex fiber, we compute completely the cohomology of the associated differential complex. We also introduce some compact models (parametrized by a certain number of forms of type (0, 1)) and, in this case, study questions of global hypoellipticity and prove, in several cases, a comparison isomorphism between the cohomology spaces. In the second part, we study (in the one dimensional case) the Fréchet algebra A^\\infty(K) (introduced recently by Cordaro, Della Sala and Lamel, to treat questions about the Borel map). We prove some function-theoretic properties about this algebra (like a localization property, in analogy with the uniform case P(K)) and study the question of determining when this algebra coincide with the full algebra of formal power series with coefficients in C(K). To do this, we introduce as the main tool a generalization of the Cauchy transform. Finally, we study a problem of regularizability of locally integrable structures. Inspired by a recent result of Kossovskiy and Zaitsev, we present a family of structures (of non-CR type) and a sufficent condition that guarantees when such structures are equivalent to real-analytic ones. |
publishDate |
2024 |
dc.date.none.fl_str_mv |
2024-02-05 |
dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
format |
doctoralThesis |
status_str |
publishedVersion |
dc.identifier.uri.fl_str_mv |
https://www.teses.usp.br/teses/disponiveis/45/45132/tde-21022024-192259/ |
url |
https://www.teses.usp.br/teses/disponiveis/45/45132/tde-21022024-192259/ |
dc.language.iso.fl_str_mv |
eng |
language |
eng |
dc.relation.none.fl_str_mv |
|
dc.rights.driver.fl_str_mv |
Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
rights_invalid_str_mv |
Liberar o conteúdo para acesso público. |
eu_rights_str_mv |
openAccess |
dc.format.none.fl_str_mv |
application/pdf |
dc.coverage.none.fl_str_mv |
|
dc.publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
publisher.none.fl_str_mv |
Biblioteca Digitais de Teses e Dissertações da USP |
dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
instname_str |
Universidade de São Paulo (USP) |
instacron_str |
USP |
institution |
USP |
reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
collection |
Biblioteca Digital de Teses e Dissertações da USP |
repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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1809090953442492416 |