One generalization of the Gagliardo inequality

Suppose $u_1,u_2,\dots,u_n\in\mathcal D(\mathbb R^k)$ and suppose we are given a certain set of linear combinations of the form $\sum_{i,j}a_{ij}^{(l)}\partial_j u_i$. Sufficient conditions in terms of the coefficients $a_{ij}^{(l)}$ are indicated for the norms $\|u_i\|_{L^{\frac k{k-1}}}$ to be controlled in terms of the $L^1$-norms these linear combinations. These conditions are most transparent if $k=2$. The classical Gagliardo inequality corresponds to a sole function $u_1=u$ and the collection of its pure partial derivatives $\partial_1 u,\dots,\partial_k u$.