This chapter introduces two sets of theorems based on a study of the Littlewood–Paley boundedness properties of a tri-linear operator. It generalizes and proves similar estimates for multi-linear multipliers whose symbols are given by characteristic functions of simplexes of arbitrary length. The chapter introduces the so-called AKNS systems, which play an important role in nuclear physics, and describes their connection with certain operators. Then, the rest of the chapter is devoted to the proof of a third theorem, which has been derived from the first two. To conclude, the chapter also proves the “delicate Bessel” Lemma which plays an important role in the argument.