AI 中文总结
该研究证明区间上带锚定的Sobolev空间的正单位球的满射等距性等价于可微阶相等，这类等距可延拓为复线性等距序同构，结论也适用于满射相位等距及1<p<∞时的满射范数可加映射。
AI 中文摘要
对于1≤p≤∞且i=1,2，设W^{k_i,p}(Ω_i)是有界开区间Ω_i上可微阶为k_i的Sobolev空间。我们为W^{k_i,p}(Ω_i)配备锚定Sobolev范数及序≥_{k_i,p}，定义为对每个j=0,…,k_i−1有f^{(j)}(x_i)≥0，且f^{(k_i)}几乎处处≥0。我们证明：当且仅当k₁=k₂时，W^{k₁,p}(Ω₁)与W^{k₂,p}(Ω₂)的正单位球是满射等距的；这类等距均可唯一延拓为复线性等距序同构，且我们给出其坐标表示。该结论对满射相位等距也成立；对1<p<∞，结论还适用于满射范数可加映射。
英文摘要
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.
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