Dominated compactness theorem in banach function spaces and its applications

Humberto Rafeiro; Stefan Samko

doi:10.1007/s11785-008-0072-z

Dominated compactness theorem in banach function spaces and its applications

Humberto Rafeiro, Stefan Samko

Research output: Contribution to journal › Article › peer-review

11 Citations (Scopus)

Abstract

A famous dominated compactness theorem due to Krasnosel'skiǐ states that compactness of a regular linear integral operator in L^p follows from that of a majorant operator. This theorem is extended to the case of the spaces Lp(ċ) (Ω, μ), μ Ω < ∞, with variable exponent p(•), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces L^p(ċ)(Ω, μ) is applied to fractional integral operators over bounded open sets.

Original language	English
Pages (from-to)	669-681
Number of pages	13
Journal	Complex Analysis and Operator Theory
Volume	2
Issue number	4
DOIs	https://doi.org/10.1007/s11785-008-0072-z
Publication status	Published - Dec 2008
Externally published	Yes

Keywords

Banach function space
Compact majorant
Compact operators
Integral operator
Krasnoselskii theorem
Potential operator
Regular operator
Variable exponent Lebesgue space

ASJC Scopus subject areas

Computational Theory and Mathematics
Computational Mathematics
Applied Mathematics

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10.1007/s11785-008-0072-z

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title = "Dominated compactness theorem in banach function spaces and its applications",

abstract = "A famous dominated compactness theorem due to Krasnosel'skiǐ states that compactness of a regular linear integral operator in Lp follows from that of a majorant operator. This theorem is extended to the case of the spaces Lp(ċ) (Ω, μ), μ Ω < ∞, with variable exponent p(•), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces Lp(ċ)(Ω, μ) is applied to fractional integral operators over bounded open sets.",

keywords = "Banach function space, Compact majorant, Compact operators, Integral operator, Krasnoselskii theorem, Potential operator, Regular operator, Variable exponent Lebesgue space",

author = "Humberto Rafeiro and Stefan Samko",

note = "Funding Information: Humberto Rafeiro gratefully acknowledges financial support by Funda¸c{\~a}o para a Ci{\^e}ncia e a Tecnologia (FCT) (Grant No. SFRH BD 22977 2005), through Programa Operacional Ci{\^e}ncia e Inova¸c{\~a}o 2010 (POCI2010) of the Portuguese Government, cofinanced by the European Community Fund FSE. Funding Information: This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792.",

year = "2008",

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doi = "10.1007/s11785-008-0072-z",

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AU - Samko, Stefan

N1 - Funding Information: Humberto Rafeiro gratefully acknowledges financial support by Funda¸cão para a Ciência e a Tecnologia (FCT) (Grant No. SFRH BD 22977 2005), through Programa Operacional Ciência e Inova¸cão 2010 (POCI2010) of the Portuguese Government, cofinanced by the European Community Fund FSE. Funding Information: This work was made under the project “Variable Exponent Analysis” supported by INTAS grant Nr.06-1000017-8792.

PY - 2008/12

Y1 - 2008/12

N2 - A famous dominated compactness theorem due to Krasnosel'skiǐ states that compactness of a regular linear integral operator in Lp follows from that of a majorant operator. This theorem is extended to the case of the spaces Lp(ċ) (Ω, μ), μ Ω < ∞, with variable exponent p(•), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces Lp(ċ)(Ω, μ) is applied to fractional integral operators over bounded open sets.

AB - A famous dominated compactness theorem due to Krasnosel'skiǐ states that compactness of a regular linear integral operator in Lp follows from that of a majorant operator. This theorem is extended to the case of the spaces Lp(ċ) (Ω, μ), μ Ω < ∞, with variable exponent p(•), where we also admit power type weights . This extension is obtained as a corollary to a more general similar dominated compactness theorem for arbitrary Banach function spaces for which the dual and associate spaces coincide. The result on compactness in the spaces Lp(ċ)(Ω, μ) is applied to fractional integral operators over bounded open sets.

KW - Banach function space

KW - Compact majorant

KW - Compact operators

KW - Integral operator

KW - Krasnoselskii theorem

KW - Potential operator

KW - Regular operator

KW - Variable exponent Lebesgue space

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JO - Complex Analysis and Operator Theory

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Dominated compactness theorem in banach function spaces and its applications

Abstract

Keywords

ASJC Scopus subject areas

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