Geodesic
One generalization of the Gagliardo inequality
Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 35, Tome 345 (2007), pp. 120-139

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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 $\partial_1 u,\dots,\partial_k u$. 
@article{ZNSL_2007_345_a7,
     author = {D. V. Maksimov},
     title = {One generalization of the {Gagliardo} inequality},
     journal = {Zapiski Nauchnykh Seminarov POMI},
     pages = {120--139},
     publisher = {mathdoc},
     volume = {345},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/ZNSL_2007_345_a7/}
}
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D. V. Maksimov. One generalization of the Gagliardo inequality. Zapiski Nauchnykh Seminarov POMI, Investigations on linear operators and function theory. Part 35, Tome 345 (2007), pp. 120-139. http://geodesic.mathdoc.fr/item/ZNSL_2007_345_a7/