Stable Liftings of Polynomial Traces on Tetrahedra

On the reference tetrahedron $K$, we construct, for each $k\in {\mathbb{N}}_0$, a right inverse for the trace operator $u\mapsto (u,{\mathrm{\partial }}_{n}u,\dots ,{\mathrm{\partial }}_n^{k}u){\mathrm{\mid }}_{\mathrm{\partial }K}$. The operator is stable as a mapping from the trace space of $W^{s,p}(K)$ to $W^{s,p}(K)$ for all $p\in (1,\mathrm{\infty })$ and $s\in (k+1\mathrm{/}p,\mathrm{\infty })$. 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.