A generalized Poincaré inequality for a class of constant coefficient differential operators

Abstract

We study first order differential operators P = P ( D ) \mathcal {P} = \mathcal {P}(D) with constant coefficients. The main question is under what conditions the following full gradient L p L^p estimate holds: \[ ‖ D ( f − f 0 ) ‖ L p ≤ C ‖ P f ‖ L p , for some f 0 ∈ ker ⁡ P . \|D(f-f_0)\|_{L^p} \leq C \|\mathcal {P} f\|_{L^p},\ \textrm {for some } f_0 \in \ker \mathcal {P}. \] We show that the constant rank condition is sufficient. The concept of the Moore-Penrose generalized inverse of a matrix comes into play.