On div-curl for higher order

Let d be an exterior derivative operator acting on differential forms on R n, defined by d: Λq(R n ) 7→ Λq+1(R n ), 0 ≤ q ≤ n. In [Math. Res. Lett. 12 (2005), no. 1, 57–61; MR2122730], E. M. Stein and the first author of the paper under review established the inequality (LS) kukLn/(n−1)(Rn) ≤ C kdukL1(Rn) + kd ∗ukL1(Rn) , which holds for any form u of degree q other than q = 1 (unless d ∗u = 0) and q = n − 1 (unless du = 0). Inequality (LS) connects the celebrated Gagliardo-Nirenberg inequality kfkLn/(n−1)(Rn) ≤ Ck∇fkL1(Rn) and the Bourgain-Brezis inequality kZkLn/(n−1)(Rn) ≤ CkCurlZkL1(Rn) for divergence-free vector fields. In the present work, the authors prove an appropriate analogue of inequality (LS) for a new class of differential operators of higher orders. {For the collection containing this paper see MR3309083}