Type and cotype of Musielak-Orlicz sequence spaces
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 \(\lambda> 0\), \(\sum_{n=1}^\infty \varphi_n (\lambda| x_n|)< \infty\). It is a Banach space under the norm \[ \| x\|_\varphi= \inf \Biggl\{ r>0: \sum_{n=1}^\infty \varphi_n \Biggl( \frac{| x_n|}{r} \Biggr)\leq 1\Biggr\}. \] 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 \(\delta^q\).
scientific article; zbMATH DE number 1228543
scientific article