An elementary approach to the homological properties of constant-rank operators

Abstract

We give a simple and constructive extension of Raiță’s result that every constant-rank operator possesses an exact potential and an exact annihilator. Our construction is completely self-contained and provides an improvement over the order of the operators constructed by Raiță and the order of the explicit annihilators for elliptic operators due to Van Schaftingen. We also give an abstract construction of an optimal annihilator for constant-rank operators, which extends the optimal construction of Van Schaftingen for elliptic operators. Lastly, we discuss the homological properties of operators in relation to the homological properties of their associated symbols. We establish that the constant-rank property is a sufficient and necessary condition for the validity of a generalized Poincaré lemma on spaces of homogeneous maps over $\mathbb{R}^d$, and we prove that the existence of potentials on spaces of periodic maps requires a strictly weaker condition than the constant-rank property.