  It is known that an elliptic system $\{P_j(x,D)\}_1^N$ of order $l$ is weakly
coercive in $\overset{\circ}{W}\rule{0pt}{2mm}^l_\infty(\mathbb R^n)$, that is,
all differential monomials of order $\le l-1$ on $C_0^\infty(\mathbb
R^n)$-functions are subordinated to this system in the $L^\infty$-norm.
Conditions for the converse result are found and other properties of weakly
coercive systems are investigated.
  An analogue of the de Leeuw-Mirkil theorem is obtained for operators with
variable coefficients: it is shown that an operator $P(x,D)$ in $n\ge 3$
variables with constant principal part is weakly coercive in
$\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb R^n)$ if and only if it is
elliptic. A similar result is obtained for systems $\{P_j(x,D)\}_1^N$ with
constant coefficients under the condition $n\ge 2N+1$ and with several
restrictions on the symbols $P_j(\xi)$ .
  A complete description of differential polynomials in two variables which are
weakly coercive in $\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb R^2)$ is
given. Wide classes of systems with constant coefficients which are weakly
coercive in $\overset{\circ}{W}\rule{0pt}{2mm}_\infty^l(\mathbb \R^n)$, but
non-elliptic are constructed.
