Metric Rigidity in Anchored Sobolev Spaces on Intervals

Min-Ruei Lin

functional analysis

Metric Rigidity in Anchored Sobolev Spaces on Intervals

arXiv:2607.27646

summary

The paper investigates when the positive unit spheres of anchored Sobolev spaces on bounded intervals are isometric, proving that this occurs only if the Sobolev orders coincide, and characterizes the resulting isometries as complex‑linear order isomorphisms with explicit coordinate forms.

Abstract

For $1 \leq p \leq \infty$ and $i = 1, 2$ , let $W^{k_{i}, p} (I^c ◯_{i})$ be the Sobolev space on a bounded open interval $I^c ◯_{i}$ with differentiability order $k_{i}$ . We equip $W^{k_{i}, p} (I^c ◯_{i})$ with an anchored Sobolev norm and the order $\geq_{k_{i}, p}$ defined by $f^{(j)} (x_{i}) \geq 0$ for each $j = 0, \dots, k_{i} - 1$ and $f^{(k_{i})} \geq 0$ a.e. We show that the positive unit spheres of $W^{k_{1}, p} (I^c ◯_{1})$ and $W^{k_{2}, p} (I^c ◯_{2})$ are surjectively isometric if and only if $k_{1} = k_{2}$ . 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 $1 < p < \infty$ , they also hold for surjective norm-additive maps.

11 pages

Topics & keywords

#sobolev spaces #anchored norm #metric rigidity #isometries #order isomorphisms #phase‑isometriesW^{k,p} spacepositive unit spheresurjective isometrynorm‑additive mapcomplex‑linear isometryinterval