Discrete Poincar\'e inequality and Discrete Trace inequality in Piece-wise Polynomial Hybridizable Spaces

In this paper, we establish discrete versions of the Poincar\'e and trace inequalities for hybridizable finite element spaces. These spaces are made of piecewise polynomial functions defined both within the interiors of elements and across all faces in a mesh's skeleton, serving as the basis for both the hybridizable discontinuous Galerkin (HDG) and hybrid high-order (HHO) methods. Additionally, we present a specific adaptation of these inequalities for the HDG method and apply them to demonstrate the stability of the related numerical schemes for second-order elliptic equations under the minimal regularity assumptions for the source term and boundary data.