Coercive estimates for linear differential operators with constant coefficients (Q1809957)

Let \(Q\) be a linear homogeneous differential operator of order \(k\) with constant coefficients and finite-dimensional kernel. The goal of this note is to construct a family of linear projection operators \(P_Q\) that satisfies \[ \begin{cases} \|P_Q u\|_{W^k_p(\Omega)}= C(\Omega,Q)\|u\|_{L^1(\Omega)},\\ \|u- P_Qu\|_{W^k_p(\Omega)}\leq C(\Omega, Q)\|Qu\|_{L_p(\Omega)}\end{cases},\tag{1} \] where \(\Omega\) is a given open set in \(\mathbb{R}^n\). To this end, the author uses the Sobolev integral representation only, which provides to obtain (1) for any homogeneous differential operator with constant coefficients provided that the kernel of the operator is finite-dimensional.