Looking for a continuous version of Bennett--Carl theorem

Abstract

We study absolute summability of inclusions of r.i. function spaces. It appears that such properties are closely related, or even determined by absolute summability of inclusions of subspaces spanned by the Rademacher system in respective r.i. spaces. Our main result states that for $1<p<2$ the inclusion $X_p\subset L^p$ is $(q,1)$-absolutely summing for each $p<q<2$, where $X_p$ is the unique r.i. Banach function space in which the Rademacher system spans copy of $l^p$. This result may be regarded as a continuous version of the well-known Carl--Bennett theorem. Two different approaches to the problem and extensive discussion on them are presented. We also conclude summability type of a kind of Sobolev embedding in the critical case.