On the compactness of paracommutators (Q1262518)

The paracommutator \[ T_ b^{st}f(\xi)=(2\pi)^{-d}\int \hat b(\xi - \eta)A(\xi,\eta)| \xi |^ s| \eta |^ tf(\eta)d\eta \] has been considered by \textit{S. Janson} and \textit{J. Peetre} [Trans. Am. Math. Soc. 305, 467-504 (1988; Zbl 0644.47046)] who, among other results, have proved results on the compactness of \(T_ b^{st}\). In this paper the author studies the converses of these results.