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Type and cotype of some Banach spaces. (English) Zbl 0787.46014
A Banach space \(X\) is said to be of cotype \((q,s)\) (resp. type \((p,s)\)), if there is a constant \(K>0\) such that \(K^{-1}(\sum^ n_{i=1} \| x_ i\|^ q)^{1/q}\leq(\int^ 1_ 0\|\sum^ n_{i=1} r_ i(t)x_ i\|^ s dt)^{1/s}\) (resp. \((\int^ 1_ 0\|\sum^ n_{i=1} r_ i(t) x_ i\|^ s dt)^{1/s}\leq K(\sum^ n_{i=1} \| x_ i\|^ p)^{1/p}\)) for every choice of \(\{x_ i\}^ n_{i=1}\subset X\) and any natural number \(n\), where \(\{r_ n\}^ \infty_{n=1}\) denotes the sequence of the Rademacher functions on \([0,1]\), \(1\leq p\leq 2\leq q<\infty\) and \(1\leq s<\infty\).
In this paper, the author mainly proves the following results. Assume that \(1<p<\infty\) and \(1\leq s<\infty\). Let \(E\) be a Banach function space and \(X= D_ E(S)=\{x\in \chi: Sx\in E\}\), where \(\chi\) is an \(F\)-space and \(S\) is a positive sublinear operator defined on \(\chi\) taking values in \(L^ 0(\Omega,\mu)\).
(1) If \(E\) is \(p\)-convex, \(1<p\leq 2\), and \(s\)-concave, and for all \(x_ 1,x_ 2,\dots,x_ n\) in \(X\),
\((\int^ 1_ 0| \sum^ n_{i=1} r_ i(t)x_ i|_ \omega^ s dt)^{1/s}\leq C_ 1(\sum^ n_{i=1} | x_ i|_ \omega^ p)^{1/p}\) \(\mu\)-a.e., then \(X\) is of type \((p,s)\).
(2) If \(E\) is \(p\)-concave, \(2\leq p<\infty\), and \(s\)-convex, and for all \(x_ 1,x_ 2,\dots,x_ n\) in \(X\), \((\sum^ n_{i=1} | x_ i|^ p_ \omega)^{1/p}\leq C_ 2(\int^ 1_ 0 |\sum^ n_{i=1} r_ i(t)x_ i|^ s_ \omega dt)^{1/s}\) \(\mu\)-a.e., then \(X\) is of cotype \((p,s)\).
Applications of the results are given.

MSC:
46B20 Geometry and structure of normed linear spaces
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
46B25 Classical Banach spaces in the general theory
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