Stable Liftings of Polynomial Traces on Tetrahedra

On the reference tetrahedron K K , we construct, for each k ∈ N₀ k ∈ N 0 , a right inverse for the trace operator u ↦ (u, ∂ ₙ u, … , ∂ ₙᵏ u)|∂ K u ↦ ( u , ∂ n u , … , ∂ n k u ) | ∂ K . The operator is stable as a mapping from the trace space of Ws, p(K) W s , p ( K ) to Ws, p(K) W s , p ( K ) for all p ∈ (1, ∞ ) p ∈ ( 1 , ∞ ) and s ∈ (k+1/p, ∞ ) s ∈ ( k + 1 / p , ∞ ) . Moreover, if the data is the trace of a polynomial of degree N ∈ N₀ N ∈ 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.