Let $$X(\mu )$$ be a p-convex ( $$1\le p<\infty $$ ) order continuous Banach function space over a positive finite measure $$\mu $$ . We characterize the subspaces of $$X(\mu )$$ which can be found simultaneously in $$X(\mu )$$ and a suitable $$L^1(\eta )$$ space, where $$\eta $$ is a positive finite measure related to the representation of $$X(\mu )$$ as an $$L^p(m)$$ space of a vector measure $$m$$ . We provide in this way new tools to analyze the strict singularity of the inclusion of $$X(\mu )$$ in such an $$L^1$$ space. No rearrangement invariant type restrictions on $$X(\mu )$$ are required.
Positivity – Springer Journals
Published: Sep 13, 2012
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