A generalization of the Gagliardo inequality

Consider functions u1, u2,..., un ∈ D(ℝk) and assume that we are given a certain set of linear combinations of the form Σi, j a ij (l) ∂jui. Sufficient conditions in terms of coefficients a ij (l) are indicated under which the norms ∥ u i∥Lk/k - 1 are controlled in terms of the L 1-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. © 2008 Springer Science+Business Media, Inc.