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Journal of Mathematical Analysis and Applications
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Journal of Mathematical Analysis and Applications
Article . 1998
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Type and Cotype of Musielak–Orlicz Sequence Spaces

Type and cotype of Musielak-Orlicz sequence spaces
Authors: Katirtzoglou, Eleni;

Type and Cotype of Musielak–Orlicz Sequence Spaces

Abstract

By a Musielak-Orlicz function \(\varphi\) is understood a sequence \((\varphi_n)\) of Young functions \(\varphi_n\). It is said to satisfy condition \(\delta^q\) for \(q\geq 1\) if there are positive constants \(k\), \(\delta\) and a nonnegative sequence \((c_n)\) in \(\ell^1\) such that for every \(n\in \mathbb{N}\), all \(x\geq 0\) and \(\lambda\geq 1\), \(\varphi_n(\lambda x)\leq k\lambda^q [\varphi_n(x)+ c_n]\) whenever \(\varphi_n(\lambda x)\leq \delta\). The Musielak-Orlicz sequence space \(\ell_\varphi\), generated by \(\varphi\), is the set of all real sequences \(x=(x_n)\) such that for some \(\lambda> 0\), \(\sum_{n=1}^\infty \varphi_n (\lambda| x_n|)0: \sum_{n=1}^\infty \varphi_n \Biggl( \frac{| x_n|}{r} \Biggr)\leq 1\Biggr\}. \] A Banach space \(X\) is said to be of cotype \(q\) if, whenever \((x_n)\) is a sequence in \(X\) such that \(\sum_{n=1}^\infty x_nr_n\) converges a.e. on \([0,1]\) (where \(r_n\) is the \(n\)th Rademacher function), then the sequence \((\| x_n\|)\) belongs to \(\ell^q\). As the main result the author proves the following statement: Let \(\varphi= (\varphi_n)\) be a Musielak-Orlicz function and \(2\leq q<\infty\). The Musielak-Orlicz sequence space \(\ell_\varphi\) is a space of cotype \(q\) if and only if the Musielak-Orlicz function \(\varphi\) satisfies the condition \(\delta^q\).

Related Organizations
Keywords

Young functions, Rademacher function, Musielak-Orlicz sequence space, Applied Mathematics, cotype \(q\), Musielak–Orlicz sequence space, type, Musielak-Orlicz function, cotype, Classical Banach spaces in the general theory, Sequence spaces (including Köthe sequence spaces), Analysis

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selected citations
These citations are derived from selected sources.
This is an alternative to the "Influence" indicator, which also reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Citations provided by BIP!
popularity
This indicator reflects the "current" impact/attention (the "hype") of an article in the research community at large, based on the underlying citation network.
BIP!Popularity provided by BIP!
influence
This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically).
BIP!Influence provided by BIP!
impulse
This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network.
BIP!Impulse provided by BIP!
18
Top 10%
Top 10%
Average
hybrid