Metric Rigidity in Anchored Sobolev Spaces on Intervals

Abstract

For $1\le p\le\infty$ 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 $\ge_{k_i,p}$ defined by $f^{(j)}(x_i)\ge 0$ for each $j=0,\ldots,k_i-1$ and $f^{(k_i)}\ge 0$ a.e. We show that the positive unit spheres of $W^{k_1,p}(Ω_1)$ and $W^{k_2,p}(Ω_2)$ 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.