
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 \(\delta^q\).
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
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
| 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). | 18 | |
| 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. | Top 10% | |
| influence This indicator reflects the overall/total impact of an article in the research community at large, based on the underlying citation network (diachronically). | Top 10% | |
| impulse This indicator reflects the initial momentum of an article directly after its publication, based on the underlying citation network. | Average |
