SYNOPSISSome characterisations of commutatitivity for C*-algebras are given in terms of inequalities involving sums and products of self-adjoint elements, and optimal constants are obtained for the corresponding inequalities for non-commutative C*-algebras.
We develop the local inequalities with new weights for the potential operator applied to differential forms. We also prove the global weighted norm inequalities for the potential operator in averaging domains and explore applications of our new results.