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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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