On the constants in inverse trace inequalities for polynomials orthogonal to lower-order subspaces

We derive sharp, explicit constants in inverse trace inequalities for polynomial functions belonging to $\mathbb{P}_p(T)$ (polynomial space with total degree $p$) that are orthogonal to the lower-order subspace $\mathbb{P}_n(T)$, $n\leq p$, where $T$ denotes a $d$-dimensional simplex. The proofs rely on orthogonal polynomial expansions on reference simplices and on a careful analysis of the eigenvalues of the relevant blocks of the face mass matrices, following the arguments developed in [9]. These results are very useful in the $hp$-analysis of the hybrid Galerkin methods, e.g. hybridizable discontinuous Galerkin methods, hybrid high-order methods, etc.