Extension of surjective isometries between unit spheres of ℓp-direct sums
Abstract
We study surjective isometries between the unit spheres of Banach spaces of the form A⊕p B, where A is nonzero and strictly convex, 1<p<∞, and every unit vector of B≠{0} has at least two distinct unit vectors at distance $2$. For two direct sums Ai⊕piBi, i=1,2, satisfying these hypotheses, we prove that every surjective isometry Δ:SA1⊕p1B1→ SA2⊕p2 B2 between their unit spheres forces equality of the direct-sum exponents pi and extends uniquely to a surjective real-linear isometry Δ̃:A1⊕p B1→ A2⊕p B2 preserving the corresponding components, where p:=p1=p2. The proof identifies the component unit spheres through diametral sets and distance formulas, and then constructs isometries between the component unit balls. As a principal application, we obtain extension and representation results for absolutely continuous functions equipped with norms combining evaluation at a fixed point and the L1-norm of the derivative. We also discuss Lipschitz and differentiable function spaces, distinguishing the consequences of our theorem from those already covered by existing extension results. Endpoint conclusions are derived separately, including the Mazur-Ulam property for the real absolutely continuous space with the sum norm.
Related
- Isometries of Unit Spheres of SFS-Spaces and a Geometric Approach to Tingley's Problem
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BibTeX
@misc{lin2026extension,
title = {{Extension of surjective isometries between unit spheres of $\ell^p$-direct sums}},
author = {Min-Ruei Lin},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.24533},
eprint = {2610.24533},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.24533},
note = {CC0},
}
RIS
TY - UNPB
TI - Extension of surjective isometries between unit spheres of ℓ^p-direct sums
AU - Min-Ruei Lin
PY - 2026
DA - 2026/10/17
PB - arXiv
DO - 10.48550/arxiv.2610.24533
UR - https://arxiv.org/abs/2610.24533
AB - We study surjective isometries between the unit spheres of Banach spaces of the form A⊕_p B, where A is nonzero and strictly convex, 1<p<∞, and every unit vector of B≠{0} has at least two distinct unit vectors at distance $2$. For two direct sums A_i⊕_p_iB_i, i=1,2, satisfying these hypotheses, we prove that every surjective isometry Δ:S_A_1⊕_p_1B_1→ S_A_2⊕_p_2 B_2 between their unit spheres forces equality of the direct-sum exponents p_i and extends uniquely to a surjective real-linear isometry Δ̃:A_1⊕_p B_1→ A_2⊕_p B_2 preserving the corresponding components, where p:=p_1=p_2. The proof identifies the component unit spheres through diametral sets and distance formulas, and then constructs isometries between the component unit balls. As a principal application, we obtain extension and representation results for absolutely continuous functions equipped with norms combining evaluation at a fixed point and the L^1-norm of the derivative. We also discuss Lipschitz and differentiable function spaces, distinguishing the consequences of our theorem from those already covered by existing extension results. Endpoint conclusions are derived separately, including the Mazur-Ulam property for the real absolutely continuous space with the sum norm.
ER -