Looking for a continuous version of Bennett--Carl theorem
Sergey V. Astashkin, Karol Leśnik, Michał Wojciechowski
TL;DR
The paper investigates absolute summability of inclusions between rearrangement invariant function spaces, revealing a strong link with the subspaces spanned by the Rademacher system. It proves that for $1<p<2$, the inclusion $X_p\subset L^p$ is $(q,1)$-absolutely summing for all $p<q<2$, using a transposition operator on mixed-norm spaces and a chain of embeddings through Orlicz-type spaces, without relying on Bennett–Carl. The authors also establish a universal weak submajorization principle to handle inclusions into $L^1$, yielding $(E,1)$-absolutely summing results for several natural target spaces such as Exp$L^q$ and $l^{p,\infty}$-type spaces, and derive Sobolev-embedding consequences. They apply these results to Sobolev spaces, showing optimal r.i. spaces for critical Sobolev embeddings and proving $(q,1)$-absolute summability for a range of Sobolev-to-Lebesgue embeddings, with a conjecture for the endpoint. Overall, the work provides a continuous analogue of the Bennett–Carl phenomenon and advances understanding of summability in functional-analytic embeddings with potential broader impact on interpolation theory and applications.
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.