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Type and cotype of Musielak-Orlicz sequence spaces. (English) Zbl 0915.46005
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 $n\in \bbfN$, 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 $\delta^q$.

46A45Sequence spaces
46B25Classical Banach spaces in the general theory of normed spaces
Full Text: DOI
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