For each $p > 1$ and each positive integer $m$, we give intrinsic characterizations of the restriction of the homogeneous Sobolev space $L_{p}^{m}(\mathbb{R})$ to an arbitrary closed subset $E$ of the real line. We show that the classical one-dimensional Whitney extension operator is "universal" for the scale of $L_{p}^{m}(\mathbb{R})$ spaces in the following sense: For every $p\in(1,\infty]$, it provides almost optimal $L^m_p$-extensions of functions defined on $E$. The operator norm of this extension operator is bounded by a constant depending only on $m$. This enables us to prove several constructive $L^m_p$-extension criteria expressed in terms of $m$-th order divided differences of functions.

中文翻译：

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一元函数的齐次Sobolev空间的扩展准则

对于每个$ p> 1 $和每个正整数$ m $，我们给出齐次Sobolev空间$ L_ {p} ^ {m}（\ mathbb {R}）$到任意封闭子集$的限制的内在表征。实线的E $。我们证明经典的一维惠特尼扩展算子在以下意义上对于$ L_ {p} ^ {m}（\ mathbb {R}）$空间的尺度是“通用的”：对于每个$ p \ in（1 ，\ infty] $，它提供了在$ E $上定义的函数的几乎最优的$ L ^ m_p $ -extensions，此扩展运算符的运算符范数受仅取决于$ m $的常数的限制，这使我们能够证明几个建设性的$ L ^ m_p $-扩展准则，以$ m $阶除以函数差异表示。