On the reference tetrahedron $K$, we construct, for each $k \in \mathbb{N}_0$, a right inverse for the trace operator $u \mapsto (u, \partial_{n} u, \ldots, \partial_{n}^k u)|_{\partial K}$. The operator is stable as a mapping from the trace space of $W^{s, p}(K)$ to $W^{s, p}(K)$ for all $p \in (1, \infty)$ and $s \in (k+1/p, \infty)$. Moreover, if the data is the trace of a polynomial of degree $N \in \mathbb{N}_0$, then the resulting lifting is a polynomial of degree $N$. One consequence of the analysis is a novel characterization for the range of the trace operator.

翻译：在参考四面体 $K$ 上，我们对每个 $k \in \mathbb{N}_0$ 构造了迹算子 $u \mapsto (u, \partial_{n} u, \ldots, \partial_{n}^k u)|_{\partial K}$ 的右逆。该算子作为从 $W^{s, p}(K)$ 的迹空间到 $W^{s, p}(K)$ 的映射，对所有 $p \in (1, \infty)$ 和 $s \in (k+1/p, \infty)$ 均具有稳定性。此外，若数据为 $N \in \mathbb{N}_0$ 次多项式的迹，则所得提升也为 $N$ 次多项式。该分析的一个推论是迹算子值域的一个新刻画。