Steklov Institute of Mathematics at St.Petersburg

PREPRINT 15/2006


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This preprint was accepted October, 2006

ABSTRACT:
Suppose $u_1, u_2, \ldots, u_n \in
\Cal D(\Bbb R^k)$ 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 $\partial_1 u, \ldots, \partial_k u$.
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