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arXiv · 2610.13833

Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces

Konstantinos Konstantos·Pavlos Motakis·Thomas Speckhofer
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Abstract

In 1981, Bourgain, Rosenthal, and Schechtman constructed uncountably many pairwise non-isomorphic complemented subspaces of $L_p$ for $1 < p < \infty$, $p \neq 2$. 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.

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Konstantinos Konstantos, Pavlos Motakis, Thomas Speckhofer. 2026-10-09. Bourgain-Rosenthal-Schechtman spaces in rearrangement-invariant and Hardy spaces. https://arxiv.org/abs/2610.13833

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