  We construct a right inverse of the trace operator $u \mapsto (u|_{\partial
T}, \partial_n u|_{\partial T})$ on the reference triangle $T$ that maps
suitable piecewise polynomial data on $\partial T$ into polynomials of the same
degree and is bounded in all $W^{s, q}(T)$ norms with $1 < q <\infty$ and $s
\geq 2$. The analysis relies on new stability estimates for three classes of
single edge operators. We then generalize the construction for $m$th-order
normal derivatives, $m \in \mathbb{N}_0$.
