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 \(\mathbb {P}_n(T)\) , \(n\leqslant p\) , 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]. The novelty is that the extremal face-mass eigenvalue is computed after removing the polynomial modes of degree at most n. This yields inverse trace inequality constants involving the factor \((p-n)(p+n+d+1)\) instead of the classical factor \((p+1)(p+d)\) , and therefore quantifies the gain in p available in projection-error estimates. 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.
