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Abstract The classical Banach space $$L_1(L_p)$$ consists of measurable scalar functions f on the unit square for which $$ \begin{align*}\|f\| = \int_0^1\Big(\int_0^1 |f(x,y)|^p dy\Big)^{1/p}dx < \infty.\end{align*} $$ We show that $$L_1(L_p) (1 < p < \infty )$$ is primary, meaning that whenever $$L_1(L_p) = E\oplus F$$ , where E and F are closed subspaces of $$L_1(L_p)$$ , then either E or F is isomorphic to $$L_1(L_p)$$ . More generally, we show that $$L_1(X)$$ is primary for a large class of rearrangement-invariant Banach function spaces.
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This content will become publicly available on March 1, 2026
On Matoušek-like embedding obstructions of countably branching graphs
In this paper we present new proofs of the non-embeddability of countably branching trees into Banach spaces satisfying property beta_p and of countably branching diamonds into Banach spaces which are l_p-asymptotic midpoint uniformly convex (p-AMUC) for p>1. These proofs are entirely metric in nature and are inspired by previous work of Jiří Matoušek. In addition, using this metric method, we succeed in extending these results to metric spaces satisfying certain embedding obstruction inequalities. Finally, we give Tessera-type lower bounds on the compression for a class of Lipschitz embeddings of the countably branching trees into Banach spaces containing l_p-asymptotic models for p>=1.
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- Award ID(s):
- 2055604
- PAR ID:
- 10625722
- Publisher / Repository:
- ELSEVIER
- Date Published:
- Journal Name:
- Journal of Mathematical Analysis and Applications
- Volume:
- 543
- Issue:
- P1
- ISSN:
- 0022-247X
- Page Range / eLocation ID:
- 128896
- Subject(s) / Keyword(s):
- Rolewicz’s property (β) Coarse embedding Non-embeddability Asymptotic midpoint uniform convexity Trees Diamonds
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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