Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces
Abstract
In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of Lp for 1 < p < ∞, p ≠ 2. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space X with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to X. To this end, we introduce an ordinal index that detects whether X embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to L1 and H1, these subspaces fail to be complemented.
Related
- The Kadets--Werner modification of Bourgain--Rosenthal space is asymptotically midpoint uniformly convex
- Characterization of the boundedness of the segment multiplier on rearrangement-invariant spaces
- Preduals of weighted homogeneous Bourgain-Morrey Besov-Triebel-Lizorkin spaces associated with operators
- Bourgain--Brezis selection maps for Hodge systems in Besov and Triebel--Lizorkin spaces
- From Sharp Self-Improvement to Exact Perimeter Formulas in Bourgain--Brezis--Mironescu Spaces
BibTeX
@misc{konstantos2026bourgainrosenthalschechtman,
title = {{Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces}},
author = {Konstantinos Konstantos and Pavlos Motakis and Thomas Speckhofer},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.13833},
eprint = {2610.13833},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.13833},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces
AU - Konstantinos Konstantos
AU - Pavlos Motakis
AU - Thomas Speckhofer
PY - 2026
DA - 2026/10/09
PB - arXiv
DO - 10.48550/arxiv.2610.13833
UR - https://arxiv.org/abs/2610.13833
AB - In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of L_p for 1 < p < ∞, p ≠ 2. We extend their construction to rearrangement-invariant spaces and Hardy spaces. For any separable rearrangement-invariant space X with non-trivial Boyd indices, we show that these subspaces are complemented, previously obtained by Ghawadrah via a different argument, and establish the following dichotomy: Either one can extract an uncountable family of pairwise non-isomorphic subspaces, or they are all isomorphic to X. To this end, we introduce an ordinal index that detects whether X embeds complementably into an arbitrary separable Banach space. On the other hand, we give sufficient conditions which imply that in certain spaces close to L_1 and H^1, these subspaces fail to be complemented.
ER -