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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Sharp affine weighted $ L^p$ Sobolev type inequalities


Authors: J. Haddad, C. H. Jiménez and M. Montenegro
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 46E35; Secondary 52A40, 52A05
DOI: https://doi.org/10.1090/tran/7728
Published electronically: December 7, 2018
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Abstract: We establish sharp affine weighted $ L^p$ Sobolev type inequalities by using the $ L_p$ Busemann-Petty centroid inequality proved by Lutwak, Yang, and Zhang. Our approach consists of combining the latter with a suitable family of sharp weighted $ L^p$ Sobolev type inequalities obtained by Nguyen and allows us to characterize all extremizers in some cases. The new inequalities do not rely on any euclidean geometric structure.


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Additional Information

J. Haddad
Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, Brazil
Email: jhaddad@mat.ufmg.br

C. H. Jiménez
Affiliation: Departamento de Matemática, Pontifícia Universidade Católica do Rio de Janeiro, Rua Marquês de São Vicente, 225 Edificio Cardeal Leme, Gávea 22.451-900, Rio de Janeiro, Brazil
Email: hugojimenez@mat.puc-rio.br

M. Montenegro
Affiliation: Departamento de Matemática, ICEx, Universidade Federal de Minas Gerais, 30.123-970 Belo Horizonte, Brazil
Email: montene@mat.ufmg.br

DOI: https://doi.org/10.1090/tran/7728
Keywords: Sobolev inequality, convex body, centroid bodies
Received by editor(s): September 27, 2017
Received by editor(s) in revised form: May 8, 2018, and July 9, 2018
Published electronically: December 7, 2018
Additional Notes: The first author was partially supported by Fapemig (APQ-01454-15).
The second author was partially supported by CNPq (PQ 305650/2016-5) and the program Incentivo à produtividade em ensino e pesquisa of the PUC-Rio.
The third author was partially supported by CNPq (PQ 306855/2016-0) and Fapemig (APQ 02574-16).
Article copyright: © Copyright 2018 American Mathematical Society