A generalization of the Gagliardo inequality

Consider functions u₁, u₂,..., u_n ∈ D(ℝ^k) and assume that we are given a certain set of linear combinations of the form ∑_{i, j} a _ij ^(l) ∂_ju_i. Sufficient conditions in terms of coefficients a _ij ^(l) are indicated under which the norms are controlled in terms of the L¹-norms of these linear combinations. These conditions are mostly transparent if k = 2. The classical Gagliardo inequality corresponds to a single function u₁ = u and the collection of its partial derivatives ∂₁u,..., ∂_ku. Bibliography: 2 titles. __________ Translated from Zapiski Nauchnykh Seminarov POMI, Vol. 345, 2007, pp. 120–139.