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Master's Dissertation
DOI
https://doi.org/10.11606/D.45.2020.tde-09092020-160421
Document
Author
Full name
Rafaela Gesing
Institute/School/College
Knowledge Area
Date of Defense
Published
São Paulo, 2020
Supervisor
Committee
Ferenczi, Valentin Raphael Henri (President)
Corrêa, Willian Hans Goes
Perez, Pedro Tradacete
Title in English
Uniform homeomorphisms between unit spheres of interpolation spaces
Keywords in English
Banach lattices
Complex interpolation
Uniform homeomorphisms between spheres
Abstract in English
This dissertation aims to present a detailed study of the article "Homéomorphismes uniformes entre les sphères unité des espaces dinterpolation" by M. Daher (1995), where he shows that, under certain hypotheses, the unit spheres of two complex interpolation spaces are uniformly homeomorphic. With this goal in mind, essential concepts will be addressed, among them, first, the theory where the results investigated are developed: theory of uniformly convex spaces, Bochner integral, and the Complex Interpolation Method of A. Calderón. Following, we present applications on the study of uniform homeomorphisms between unit spheres of Banach spaces on the interpolation scale, including the context of Lp spaces and weighted Lp spaces. Finally, we introduce some topics on the theory of Banach lattices and its interplay with interpolation theory, presenting the Calderón-Lozanovskii construction and the uniform homeomorphism between unit spheres in this setting.
Title in Portuguese
Homeomorfismos uniformes entre esferas unitárias de espaços de interpolação
Keywords in Portuguese
Homeomorfismos uniformes entre esferas
Interpolação complexa
Reticulados de Banach
Abstract in Portuguese
Esta dissertação tem por objetivo apresentar um estudo detalhado do artigo "Homéomorphismes uniformes entre les sphères unité des espaces dinterpolation" de M. Daher (1995), onde ele mostra que, sob certas hipóteses, as esferas unitárias de dois espaços de interpolação complexa são uniformemente homeomorfas. Com esse intuito em mente, conceitos essenciais serão abordados, entre eles, primeiramente, a teoria em que os resultados investigados são desenvolvidos: teoria de espaços uniformemente convexos, integral de Bochner, Método Complexo de Interpolação de A. Calderón. Em segundo lugar, apresentamos aplicações do estudo de homeomorfismos uniformes entre esferas unitárias da escala de interpolação de espaços de Banach, incluindo os espaços Lp e os espaços Lp com peso. Finalmente, introduzimos alguns tópicos sobre a teoria de reticulados de Banach e sua interação com a teoria de interpolação, apresentando a construção de Calderón-Lozanovskii e o homeomorfismo uniforme entre as esferas unitárias nesse cenário.
 
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Dissertacao__RG.pdf (1.04 Mbytes)
Publishing Date
2021-01-20
 
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