A Class of Counterexamples to the Fefferman–Phong Inequality for Systems.

We give here a class of counterexamples to the Fefferman–Phong inequality for systems of pseudodifferential operators, which contains Brummelhuis’ one as a particular case. The main ingredient in the proof is the use of“localized operators” associated with the system, and Hörmander's example of a positive-semidefinite matrix whose Weyl quantization is not nonnegative. For the considered class, in the“isotropic” case, the Sharp Gårding inequality cannot be improved. [ABSTRACT FROM AUTHOR]