This chapter produces a new class of differential operators of order k (where k is any given positive integer) that satisfy an appropriate analogue of a Gagliardo–Nirenberg inequality for functions and contain the operators introduced in the works of Bourgain and Brezis and Van Schaftingen. The research in this chapter is furthermore based on div/curl-type phenomena studied by both Stein as well as one of the authors of this chapter. Thus, the chapter first introduces the notion of admissible degree increment, and describes the necessary operators and theorems. Proofs are then discussed later on in the chapter, before it concludes with further remarks on some of the problems and theorems advanced earlier on.