Metric Rigidity in Anchored Sobolev Spaces on Intervals

2026/07/30 by Min-Ruei Lin
Mathematics · #math.FA

paper · pdf

11 pages

arxiv created 2026/07/30 · arxiv updated 2026/07/31

Abstract

For 1≤ p≤∞ and i=1,2, let W^k_i,p(Ω_i) be the Sobolev space on a bounded open interval Ω_i with differentiability order k_i. We equip W^k_i,p(Ω_i) with an anchored Sobolev norm and the order ≥_ki,p defined by f^(j)(x_i)≥ 0 for each j=0,…,k_i-1 and f^(k_i)≥ 0 a.e. We show that the positive unit spheres of W^k₁,p(Ω₁) and W^k₂,p(Ω₂) are surjectively isometric if and only if k₁=k₂. Every such isometry extends uniquely to a complex-linear isometric order isomorphism, for which we obtain a coordinate representation. The same conclusions hold for surjective phase-isometries. For 1<p<∞, they also hold for surjective norm-additive maps.

Citations

• The Mazur-Ulam property for a Banach space which satisfies a separation condition
• Exploring new solutions to Tingley's problem for function algebras
• The Mazur-Ulam property for uniform algebras
• On a variant of Tingley's problem for some function spaces
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